diff --git a/thys/Epistemic_Logic/Epistemic_Logic.thy b/thys/Epistemic_Logic/Epistemic_Logic.thy
--- a/thys/Epistemic_Logic/Epistemic_Logic.thy
+++ b/thys/Epistemic_Logic/Epistemic_Logic.thy
@@ -1,1605 +1,1411 @@
 (*
   File:      Epistemic_Logic.thy
   Author:    Asta Halkjær From
 
   This work is a formalization of epistemic logic with countably many agents.
   It includes proofs of soundness and completeness for the axiom system K.
   The completeness proof is based on the textbook "Reasoning About Knowledge"
   by Fagin, Halpern, Moses and Vardi (MIT Press 1995).
   The extensions of system K (T, KB, K4, S4, S5) and their completeness proofs
   are based on the textbook "Modal Logic" by Blackburn, de Rijke and Venema
   (Cambridge University Press 2001).
 *)
 
 theory Epistemic_Logic imports "HOL-Library.Countable" begin
 
 section \<open>Syntax\<close>
 
 type_synonym id = string
 
 datatype 'i fm
   = FF ("\<^bold>\<bottom>")
   | Pro id
   | Dis \<open>'i fm\<close> \<open>'i fm\<close> (infixr "\<^bold>\<or>" 30)
   | Con \<open>'i fm\<close> \<open>'i fm\<close> (infixr "\<^bold>\<and>" 35)
   | Imp \<open>'i fm\<close> \<open>'i fm\<close> (infixr "\<^bold>\<longrightarrow>" 25)
   | K 'i \<open>'i fm\<close>
 
 abbreviation TT ("\<^bold>\<top>") where
   \<open>TT \<equiv> \<^bold>\<bottom> \<^bold>\<longrightarrow> \<^bold>\<bottom>\<close>
 
 abbreviation Neg ("\<^bold>\<not> _" [40] 40) where
   \<open>Neg p \<equiv> p \<^bold>\<longrightarrow> \<^bold>\<bottom>\<close>
 
 abbreviation \<open>L i p \<equiv> \<^bold>\<not> K i (\<^bold>\<not> p)\<close>
 
 section \<open>Semantics\<close>
 
-datatype ('i, 'w) kripke = Kripke (\<pi>: \<open>'w \<Rightarrow> id \<Rightarrow> bool\<close>) (\<K>: \<open>'i \<Rightarrow> 'w \<Rightarrow> 'w set\<close>)
+datatype ('i, 'w) kripke =
+  Kripke (\<W>: \<open>'w set\<close>) (\<pi>: \<open>'w \<Rightarrow> id \<Rightarrow> bool\<close>) (\<K>: \<open>'i \<Rightarrow> 'w \<Rightarrow> 'w set\<close>)
 
 primrec semantics :: \<open>('i, 'w) kripke \<Rightarrow> 'w \<Rightarrow> 'i fm \<Rightarrow> bool\<close>
   ("_, _ \<Turnstile> _" [50, 50] 50) where
   \<open>(M, w \<Turnstile> \<^bold>\<bottom>) = False\<close>
 | \<open>(M, w \<Turnstile> Pro x) = \<pi> M w x\<close>
 | \<open>(M, w \<Turnstile> (p \<^bold>\<or> q)) = ((M, w \<Turnstile> p) \<or> (M, w \<Turnstile> q))\<close>
 | \<open>(M, w \<Turnstile> (p \<^bold>\<and> q)) = ((M, w \<Turnstile> p) \<and> (M, w \<Turnstile> q))\<close>
 | \<open>(M, w \<Turnstile> (p \<^bold>\<longrightarrow> q)) = ((M, w \<Turnstile> p) \<longrightarrow> (M, w \<Turnstile> q))\<close>
-| \<open>(M, w \<Turnstile> K i p) = (\<forall>v \<in> \<K> M i w. M, v \<Turnstile> p)\<close>
+| \<open>(M, w \<Turnstile> K i p) = (\<forall>v \<in> \<W> M \<inter> \<K> M i w. M, v \<Turnstile> p)\<close>
 
 section \<open>S5 Axioms\<close>
 
 definition reflexive :: \<open>('i, 'w) kripke \<Rightarrow> bool\<close> where
-  \<open>reflexive M \<equiv> \<forall>i w. w \<in> \<K> M i w\<close>
+  \<open>reflexive M \<equiv> \<forall>i. \<forall>w \<in> \<W> M. w \<in> \<K> M i w\<close>
 
 definition symmetric :: \<open>('i, 'w) kripke \<Rightarrow> bool\<close> where
-  \<open>symmetric M \<equiv> \<forall>i v w. v \<in> \<K> M i w \<longleftrightarrow> w \<in> \<K> M i v\<close>
+  \<open>symmetric M \<equiv> \<forall>i. \<forall>v \<in> \<W> M. \<forall>w \<in> \<W> M. v \<in> \<K> M i w \<longleftrightarrow> w \<in> \<K> M i v\<close>
 
 definition transitive :: \<open>('i, 'w) kripke \<Rightarrow> bool\<close> where
-  \<open>transitive M \<equiv> \<forall>i v w u. w \<in> \<K> M i v \<and> u \<in> \<K> M i w \<longrightarrow> u \<in> \<K> M i v\<close>
+  \<open>transitive M \<equiv> \<forall>i. \<forall>u \<in> \<W> M. \<forall>v \<in> \<W> M. \<forall>w \<in> \<W> M.
+    w \<in> \<K> M i v \<and> u \<in> \<K> M i w \<longrightarrow> u \<in> \<K> M i v\<close>
 
 abbreviation equivalence :: \<open>('i, 'w) kripke \<Rightarrow> bool\<close> where
   \<open>equivalence M \<equiv> reflexive M \<and> symmetric M \<and> transitive M\<close>
 
 lemma Imp_intro [intro]: \<open>(M, w \<Turnstile> p \<Longrightarrow> M, w \<Turnstile> q) \<Longrightarrow> M, w \<Turnstile> (p \<^bold>\<longrightarrow> q)\<close>
   by simp
 
 theorem distribution: \<open>M, w \<Turnstile> (K i p \<^bold>\<and> K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i q)\<close>
 proof
   assume \<open>M, w \<Turnstile> (K i p \<^bold>\<and> K i (p \<^bold>\<longrightarrow> q))\<close>
   then have \<open>M, w \<Turnstile> K i p\<close> \<open>M, w \<Turnstile> K i (p \<^bold>\<longrightarrow> q)\<close>
     by simp_all
-  then have \<open>\<forall>v \<in> \<K> M i w. M, v \<Turnstile> p\<close> \<open>\<forall>v \<in> \<K> M i w. M, v \<Turnstile> (p \<^bold>\<longrightarrow> q)\<close>
+  then have \<open>\<forall>v \<in> \<W> M \<inter> \<K> M i w. M, v \<Turnstile> p\<close> \<open>\<forall>v \<in> \<W> M \<inter> \<K> M i w. M, v \<Turnstile> (p \<^bold>\<longrightarrow> q)\<close>
     by simp_all
-  then have \<open>\<forall>v \<in> \<K> M i w. M, v \<Turnstile> q\<close>
+  then have \<open>\<forall>v \<in> \<W> M \<inter> \<K> M i w. M, v \<Turnstile> q\<close>
     by simp
   then show \<open>M, w \<Turnstile> K i q\<close>
     by simp
 qed
 
 theorem generalization:
   assumes valid: \<open>\<forall>(M :: ('i, 'w) kripke) w. M, w \<Turnstile> p\<close>
   shows \<open>(M :: ('i, 'w) kripke), w \<Turnstile> K i p\<close>
 proof -
   have \<open>\<forall>w' \<in> \<K> M i w. M, w' \<Turnstile> p\<close>
     using valid by blast
   then show \<open>M, w \<Turnstile> K i p\<close>
     by simp
 qed
 
 theorem truth:
-  assumes \<open>reflexive M\<close>
+  assumes \<open>reflexive M\<close> \<open>w \<in> \<W> M\<close>
   shows \<open>M, w \<Turnstile> (K i p \<^bold>\<longrightarrow> p)\<close>
 proof
   assume \<open>M, w \<Turnstile> K i p\<close>
-  then have \<open>\<forall>v \<in> \<K> M i w. M, v \<Turnstile> p\<close>
+  then have \<open>\<forall>v \<in> \<W> M \<inter> \<K> M i w. M, v \<Turnstile> p\<close>
     by simp
   moreover have \<open>w \<in> \<K> M i w\<close>
-    using \<open>reflexive M\<close> unfolding reflexive_def by blast
+    using \<open>reflexive M\<close> \<open>w \<in> \<W> M\<close> unfolding reflexive_def by blast
   ultimately show \<open>M, w \<Turnstile> p\<close>
-    by simp
+    using \<open>w \<in> \<W> M\<close> by simp
 qed
 
 theorem pos_introspection:
-  assumes \<open>transitive M\<close>
+  assumes \<open>transitive M\<close> \<open>w \<in> \<W> M\<close>
   shows \<open>M, w \<Turnstile> (K i p \<^bold>\<longrightarrow> K i (K i p))\<close>
 proof
   assume \<open>M, w \<Turnstile> K i p\<close>
-  then have \<open>\<forall>v \<in> \<K> M i w. M, v \<Turnstile> p\<close>
+  then have \<open>\<forall>v \<in> \<W> M \<inter> \<K> M i w. M, v \<Turnstile> p\<close>
     by simp
-  then have \<open>\<forall>v \<in> \<K> M i w. \<forall>u \<in> \<K> M i v. M, u \<Turnstile> p\<close>
-    using \<open>transitive M\<close> unfolding transitive_def by blast
-  then have \<open>\<forall>v \<in> \<K> M i w. M, v \<Turnstile> K i p\<close>
+  then have \<open>\<forall>v \<in> \<W> M \<inter> \<K> M i w. \<forall>u \<in> \<W> M \<inter> \<K> M i v. M, u \<Turnstile> p\<close>
+    using \<open>transitive M\<close> \<open>w \<in> \<W> M\<close> unfolding transitive_def by blast
+  then have \<open>\<forall>v \<in> \<W> M \<inter> \<K> M i w. M, v \<Turnstile> K i p\<close>
     by simp
   then show \<open>M, w \<Turnstile> K i (K i p)\<close>
     by simp
 qed
 
 theorem neg_introspection:
-  assumes \<open>symmetric M\<close> \<open>transitive M\<close>
+  assumes \<open>symmetric M\<close> \<open>transitive M\<close> \<open>w \<in> \<W> M\<close>
   shows \<open>M, w \<Turnstile> (\<^bold>\<not> K i p \<^bold>\<longrightarrow> K i (\<^bold>\<not> K i p))\<close>
 proof
   assume \<open>M, w \<Turnstile> \<^bold>\<not> (K i p)\<close>
-  then obtain u where \<open>u \<in> \<K> M i w\<close> \<open>\<not> (M, u \<Turnstile> p)\<close>
+  then obtain u where \<open>u \<in> \<K> M i w\<close> \<open>\<not> (M, u \<Turnstile> p)\<close> \<open>u \<in> \<W> M\<close>
     by auto
-  moreover have \<open>\<forall>v \<in> \<K> M i w. u \<in> \<K> M i v\<close>
-    using \<open>u \<in> \<K> M i w\<close> \<open>symmetric M\<close> \<open>transitive M\<close>
+  moreover have \<open>\<forall>v \<in> \<W> M \<inter> \<K> M i w. u \<in> \<W> M \<inter> \<K> M i v\<close>
+    using \<open>u \<in> \<K> M i w\<close> \<open>symmetric M\<close> \<open>transitive M\<close> \<open>u \<in> \<W> M\<close> \<open>w \<in> \<W> M\<close>
     unfolding symmetric_def transitive_def by blast
-  ultimately have \<open>\<forall>v \<in> \<K> M i w. M, v \<Turnstile> \<^bold>\<not> K i p\<close>
+  ultimately have \<open>\<forall>v \<in> \<W> M \<inter> \<K> M i w. M, v \<Turnstile> \<^bold>\<not> K i p\<close>
     by auto
   then show \<open>M, w \<Turnstile> K i (\<^bold>\<not> K i p)\<close>
     by simp
 qed
 
-lemma ex_equiv_model: \<open>\<exists>M. reflexive M \<and> symmetric M \<and> transitive M\<close>
-proof (intro exI conjI)
-  let ?M = \<open>(Kripke undefined (\<lambda>_ w. {w}))\<close>
-  show \<open>reflexive ?M\<close>
-    unfolding reflexive_def by simp
-  show \<open>symmetric ?M\<close>
-    unfolding symmetric_def by auto
-  show \<open>transitive ?M\<close>
-    unfolding transitive_def by simp
-qed
-
 section \<open>Normal Modal Logic\<close>
 
 primrec eval :: \<open>(id \<Rightarrow> bool) \<Rightarrow> ('i fm \<Rightarrow> bool) \<Rightarrow> 'i fm \<Rightarrow> bool\<close> where
   \<open>eval _ _ \<^bold>\<bottom> = False\<close>
 | \<open>eval g _ (Pro x) = g x\<close>
 | \<open>eval g h (p \<^bold>\<or> q) = (eval g h p \<or> eval g h q)\<close>
 | \<open>eval g h (p \<^bold>\<and> q) = (eval g h p \<and> eval g h q)\<close>
 | \<open>eval g h (p \<^bold>\<longrightarrow> q) = (eval g h p \<longrightarrow> eval g h q)\<close>
 | \<open>eval _ h (K i p) = h (K i p)\<close>
 
 abbreviation \<open>tautology p \<equiv> \<forall>g h. eval g h p\<close>
 
 inductive AK :: \<open>('i fm \<Rightarrow> bool) \<Rightarrow> 'i fm \<Rightarrow> bool\<close> ("_ \<turnstile> _" [50, 50] 50)
   for A :: \<open>'i fm \<Rightarrow> bool\<close> where
     A1: \<open>tautology p \<Longrightarrow> A \<turnstile> p\<close>
   | A2: \<open>A \<turnstile> (K i p \<^bold>\<and> K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i q)\<close>
   | Ax: \<open>A p \<Longrightarrow> A \<turnstile> p\<close>
   | R1: \<open>A \<turnstile> p \<Longrightarrow> A \<turnstile> (p \<^bold>\<longrightarrow> q) \<Longrightarrow> A \<turnstile> q\<close>
   | R2: \<open>A \<turnstile> p \<Longrightarrow> A \<turnstile> K i p\<close>
 
 section \<open>Soundness\<close>
 
-lemma eval_semantics: \<open>eval (pi w) (\<lambda>q. Kripke pi r, w \<Turnstile> q) p = (Kripke pi r, w \<Turnstile> p)\<close>
+lemma eval_semantics: \<open>eval (pi w) (\<lambda>q. Kripke W pi r, w \<Turnstile> q) p = (Kripke W pi r, w \<Turnstile> p)\<close>
   by (induct p) simp_all
 
 lemma tautology:
   assumes \<open>tautology p\<close>
   shows \<open>M, w \<Turnstile> p\<close>
 proof -
-  from assms have \<open>eval (g w) (\<lambda>q. Kripke g r, w \<Turnstile> q) p\<close> for g r
+  from assms have \<open>eval (g w) (\<lambda>q. Kripke W g r, w \<Turnstile> q) p\<close> for W g r
     by simp
-  then have \<open>Kripke g r, w \<Turnstile> p\<close> for g r
+  then have \<open>Kripke W g r, w \<Turnstile> p\<close> for W g r
     using eval_semantics by fast
   then show \<open>M, w \<Turnstile> p\<close>
     by (metis kripke.collapse)
 qed
 
 theorem soundness:
   fixes M :: \<open>('i, 'w) kripke\<close>
-  assumes \<open>\<And>(M :: ('i, 'w) kripke) w p. A p \<Longrightarrow> P M \<Longrightarrow> M, w \<Turnstile> p\<close>
-  shows \<open>A \<turnstile> p \<Longrightarrow> P M \<Longrightarrow> M, w \<Turnstile> p\<close>
-  by (induct p arbitrary: w rule: AK.induct) (simp_all add: assms tautology)
+  assumes \<open>\<And>(M :: ('i, 'w) kripke) w p. A p \<Longrightarrow> P M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
+  shows \<open>A \<turnstile> p \<Longrightarrow> P M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
+  by (induct p arbitrary: w rule: AK.induct) (auto simp: assms tautology)
 
 section \<open>Derived rules\<close>
 
 lemma K_A2': \<open>A \<turnstile> (K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i p \<^bold>\<longrightarrow> K i q)\<close>
 proof -
   have \<open>A \<turnstile> (K i p \<^bold>\<and> K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i q)\<close>
     using A2 by fast
   moreover have \<open>A \<turnstile> ((P \<^bold>\<and> Q \<^bold>\<longrightarrow> R) \<^bold>\<longrightarrow> (Q \<^bold>\<longrightarrow> P \<^bold>\<longrightarrow> R))\<close> for P Q R
     by (simp add: A1)
   ultimately show ?thesis
     using R1 by fast
 qed
 
 lemma K_map:
   assumes \<open>A \<turnstile> (p \<^bold>\<longrightarrow> q)\<close>
   shows \<open>A \<turnstile> (K i p \<^bold>\<longrightarrow> K i q)\<close>
 proof -
   note \<open>A \<turnstile> (p \<^bold>\<longrightarrow> q)\<close>
   then have \<open>A \<turnstile> K i (p \<^bold>\<longrightarrow> q)\<close>
     using R2 by fast
   moreover have \<open>A \<turnstile> (K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i p \<^bold>\<longrightarrow> K i q)\<close>
     using K_A2' by fast
   ultimately show ?thesis
     using R1 by fast
 qed
 
 lemma K_LK: \<open>A \<turnstile> (L i (\<^bold>\<not> p) \<^bold>\<longrightarrow> \<^bold>\<not> K i p)\<close>
 proof -
   have \<open>A \<turnstile> (p \<^bold>\<longrightarrow> \<^bold>\<not> \<^bold>\<not> p)\<close>
     by (simp add: A1)
   moreover have \<open>A \<turnstile> ((P \<^bold>\<longrightarrow> Q) \<^bold>\<longrightarrow> (\<^bold>\<not> Q \<^bold>\<longrightarrow> \<^bold>\<not> P))\<close> for P Q
     using A1 by force
   ultimately show ?thesis
     using K_map R1 by fast
 qed
 
 primrec imply :: \<open>'i fm list \<Rightarrow> 'i fm \<Rightarrow> 'i fm\<close> where
   \<open>imply [] q = q\<close>
 | \<open>imply (p # ps) q = (p \<^bold>\<longrightarrow> imply ps q)\<close>
 
 lemma K_imply_head: \<open>A \<turnstile> imply (p # ps) p\<close>
 proof -
   have \<open>tautology (imply (p # ps) p)\<close>
     by (induct ps) simp_all
   then show ?thesis
     using A1 by blast
 qed
 
 lemma K_imply_Cons:
   assumes \<open>A \<turnstile> imply ps q\<close>
   shows \<open>A \<turnstile> imply (p # ps) q\<close>
 proof -
   have \<open>A \<turnstile> (imply ps q \<^bold>\<longrightarrow> imply (p # ps) q)\<close>
     by (simp add: A1)
   with R1 assms show ?thesis .
 qed
 
 lemma K_right_mp:
   assumes \<open>A \<turnstile> imply ps p\<close> \<open>A \<turnstile> imply ps (p \<^bold>\<longrightarrow> q)\<close>
   shows \<open>A \<turnstile> imply ps q\<close>
 proof -
   have \<open>tautology (imply ps p \<^bold>\<longrightarrow> imply ps (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> imply ps q)\<close>
     by (induct ps) simp_all
   with A1 have \<open>A \<turnstile> (imply ps p \<^bold>\<longrightarrow> imply ps (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> imply ps q)\<close> .
   then show ?thesis
     using assms R1 by blast
 qed
 
 lemma tautology_imply_superset:
   assumes \<open>set ps \<subseteq> set qs\<close>
   shows \<open>tautology (imply ps r \<^bold>\<longrightarrow> imply qs r)\<close>
 proof (rule ccontr)
   assume \<open>\<not> tautology (imply ps r \<^bold>\<longrightarrow> imply qs r)\<close>
   then obtain g h where \<open>\<not> eval g h (imply ps r \<^bold>\<longrightarrow> imply qs r)\<close>
     by blast
   then have \<open>eval g h (imply ps r)\<close> \<open>\<not> eval g h (imply qs r)\<close>
     by simp_all
   then consider (np) \<open>\<exists>p \<in> set ps. \<not> eval g h p\<close> | (r) \<open>\<forall>p \<in> set ps. eval g h p\<close> \<open>eval g h r\<close>
     by (induct ps) auto
   then show False
   proof cases
     case np
     then have \<open>\<exists>p \<in> set qs. \<not> eval g h p\<close>
       using \<open>set ps \<subseteq> set qs\<close> by blast
     then have \<open>eval g h (imply qs r)\<close>
       by (induct qs) simp_all
     then show ?thesis
       using \<open>\<not> eval g h (imply qs r)\<close> by blast
   next
     case r
     then have \<open>eval g h (imply qs r)\<close>
       by (induct qs) simp_all
     then show ?thesis
       using \<open>\<not> eval g h (imply qs r)\<close> by blast
   qed
 qed
 
 lemma K_imply_weaken:
   assumes \<open>A \<turnstile> imply ps q\<close> \<open>set ps \<subseteq> set ps'\<close>
   shows \<open>A \<turnstile> imply ps' q\<close>
 proof -
   have \<open>tautology (imply ps q \<^bold>\<longrightarrow> imply ps' q)\<close>
     using \<open>set ps \<subseteq> set ps'\<close> tautology_imply_superset by blast
   then have \<open>A \<turnstile> (imply ps q \<^bold>\<longrightarrow> imply ps' q)\<close>
     using A1 by blast
   then show ?thesis
     using \<open>A \<turnstile> imply ps q\<close> R1 by blast
 qed
 
 lemma imply_append: \<open>imply (ps @ ps') q = imply ps (imply ps' q)\<close>
   by (induct ps) simp_all
 
 lemma K_ImpI:
   assumes \<open>A \<turnstile> imply (p # G) q\<close>
   shows \<open>A \<turnstile> imply G (p \<^bold>\<longrightarrow> q)\<close>
 proof -
   have \<open>set (p # G) \<subseteq> set (G @ [p])\<close>
     by simp
   then have \<open>A \<turnstile> imply (G @ [p]) q\<close>
     using assms K_imply_weaken by blast
   then have \<open>A \<turnstile> imply G (imply [p] q)\<close>
     using imply_append by metis
   then show ?thesis
     by simp
 qed
 
 lemma K_Boole:
   assumes \<open>A \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close>
   shows \<open>A \<turnstile> imply G p\<close>
 proof -
   have \<open>A \<turnstile> imply G (\<^bold>\<not> \<^bold>\<not> p)\<close>
     using assms K_ImpI by blast
   moreover have \<open>tautology (imply G (\<^bold>\<not> \<^bold>\<not> p) \<^bold>\<longrightarrow> imply G p)\<close>
     by (induct G) simp_all
   then have \<open>A \<turnstile> (imply G (\<^bold>\<not> \<^bold>\<not> p) \<^bold>\<longrightarrow> imply G p)\<close>
     using A1 by blast
   ultimately show ?thesis
     using R1 by blast
 qed
 
 lemma K_DisE:
   assumes \<open>A \<turnstile> imply (p # G) r\<close> \<open>A \<turnstile> imply (q # G) r\<close> \<open>A \<turnstile> imply G (p \<^bold>\<or> q)\<close>
   shows \<open>A \<turnstile> imply G r\<close>
 proof -
   have \<open>tautology (imply (p # G) r \<^bold>\<longrightarrow> imply (q # G) r \<^bold>\<longrightarrow> imply G (p \<^bold>\<or> q) \<^bold>\<longrightarrow> imply G r)\<close>
     by (induct G) auto
   then have \<open>A \<turnstile> (imply (p # G) r \<^bold>\<longrightarrow> imply (q # G) r \<^bold>\<longrightarrow> imply G (p \<^bold>\<or> q) \<^bold>\<longrightarrow> imply G r)\<close>
     using A1 by blast
   then show ?thesis
     using assms R1 by blast
 qed
 
 lemma K_mp: \<open>A \<turnstile> imply (p # (p \<^bold>\<longrightarrow> q) # G) q\<close>
   by (meson K_imply_head K_imply_weaken K_right_mp set_subset_Cons)
 
 lemma K_swap:
   assumes \<open>A \<turnstile> imply (p # q # G) r\<close>
   shows \<open>A \<turnstile> imply (q # p # G) r\<close>
   using assms K_ImpI by (metis imply.simps(1-2))
 
 lemma K_DisL:
   assumes \<open>A \<turnstile> imply (p # ps) q\<close> \<open>A \<turnstile> imply (p' # ps) q\<close>
   shows \<open>A \<turnstile> imply ((p \<^bold>\<or> p') # ps) q\<close>
 proof -
   have \<open>A \<turnstile> imply (p # (p \<^bold>\<or> p') # ps) q\<close> \<open>A \<turnstile> imply (p' # (p \<^bold>\<or> p') # ps) q\<close>
     using assms K_swap K_imply_Cons by blast+
   moreover have \<open>A \<turnstile> imply ((p \<^bold>\<or> p') # ps) (p \<^bold>\<or> p')\<close>
     using K_imply_head by blast
   ultimately show ?thesis
     using K_DisE by blast
 qed
 
 lemma K_distrib_K_imp:
   assumes \<open>A \<turnstile> K i (imply G q)\<close>
   shows \<open>A \<turnstile> imply (map (K i) G) (K i q)\<close>
 proof -
   have \<open>A \<turnstile> (K i (imply G q) \<^bold>\<longrightarrow> imply (map (K i) G) (K i q))\<close>
   proof (induct G)
     case Nil
     then show ?case
       by (simp add: A1)
   next
     case (Cons a G)
     have \<open>A \<turnstile> (K i a \<^bold>\<and> K i (imply (a # G) q) \<^bold>\<longrightarrow> K i (imply G q))\<close>
       by (simp add: A2)
     moreover have
       \<open>A \<turnstile> ((K i a \<^bold>\<and> K i (imply (a # G) q) \<^bold>\<longrightarrow> K i (imply G q)) \<^bold>\<longrightarrow>
         (K i (imply G q) \<^bold>\<longrightarrow> imply (map (K i) G) (K i q)) \<^bold>\<longrightarrow>
         (K i a \<^bold>\<and> K i (imply (a # G) q) \<^bold>\<longrightarrow> imply (map (K i) G) (K i q)))\<close>
       by (simp add: A1)
     ultimately have \<open>A \<turnstile> (K i a \<^bold>\<and> K i (imply (a # G) q) \<^bold>\<longrightarrow> imply (map (K i) G) (K i q))\<close>
       using Cons R1 by blast
     moreover have
       \<open>A \<turnstile> ((K i a \<^bold>\<and> K i (imply (a # G) q) \<^bold>\<longrightarrow> imply (map (K i) G) (K i q)) \<^bold>\<longrightarrow>
         (K i (imply (a # G) q) \<^bold>\<longrightarrow> K i a \<^bold>\<longrightarrow> imply (map (K i) G) (K i q)))\<close>
       by (simp add: A1)
     ultimately have \<open>A \<turnstile> (K i (imply (a # G) q) \<^bold>\<longrightarrow> K i a \<^bold>\<longrightarrow> imply (map (K i) G) (K i q))\<close>
       using R1 by blast
     then show ?case
       by simp
   qed
   then show ?thesis
     using assms R1 by blast
 qed
 
 section \<open>Completeness\<close>
 
 subsection \<open>Consistent sets\<close>
 
 definition consistent :: \<open>('i fm \<Rightarrow> bool) \<Rightarrow> 'i fm set \<Rightarrow> bool\<close> where
   \<open>consistent A S \<equiv> \<nexists>S'. set S' \<subseteq> S \<and> A \<turnstile> imply S' \<^bold>\<bottom>\<close>
 
 lemma inconsistent_subset:
   assumes \<open>consistent A V\<close> \<open>\<not> consistent A ({p} \<union> V)\<close>
   obtains V' where \<open>set V' \<subseteq> V\<close> \<open>A \<turnstile> imply (p # V') \<^bold>\<bottom>\<close>
 proof -
   obtain V' where V': \<open>set V' \<subseteq> ({p} \<union> V)\<close> \<open>p \<in> set V'\<close> \<open>A \<turnstile> imply V' \<^bold>\<bottom>\<close>
     using assms unfolding consistent_def by blast
   then have *: \<open>A \<turnstile> imply (p # V') \<^bold>\<bottom>\<close>
     using K_imply_Cons by blast
 
   let ?S = \<open>removeAll p V'\<close>
   have \<open>set (p # V') \<subseteq> set (p # ?S)\<close>
     by auto
   then have \<open>A \<turnstile> imply (p # ?S) \<^bold>\<bottom>\<close>
     using * K_imply_weaken by blast
   moreover have \<open>set ?S \<subseteq> V\<close>
     using V'(1) by (metis Diff_subset_conv set_removeAll)
   ultimately show ?thesis
     using that by blast
 qed
 
 lemma consistent_deriv:
   assumes \<open>consistent A V\<close> \<open>A \<turnstile> p\<close>
   shows \<open>consistent A ({p} \<union> V)\<close>
   using assms by (metis R1 consistent_def imply.simps(2) inconsistent_subset)
 
 lemma consistent_consequent:
   assumes \<open>consistent A V\<close> \<open>p \<in> V\<close> \<open>A \<turnstile> (p \<^bold>\<longrightarrow> q)\<close>
   shows \<open>consistent A ({q} \<union> V)\<close>
 proof -
   have \<open>\<forall>V'. set V' \<subseteq> V \<longrightarrow> \<not> A \<turnstile> imply (p # V') \<^bold>\<bottom>\<close>
     using \<open>consistent A V\<close> \<open>p \<in> V\<close> unfolding consistent_def
     by (metis insert_subset list.simps(15))
   then have \<open>\<forall>V'. set V' \<subseteq> V \<longrightarrow> \<not> A \<turnstile> imply (q # V') \<^bold>\<bottom>\<close>
     using \<open>A \<turnstile> (p \<^bold>\<longrightarrow> q)\<close> K_imply_head K_right_mp by (metis imply.simps(1-2))
   then show ?thesis
     using \<open>consistent A V\<close> inconsistent_subset by metis
 qed
 
 lemma consistent_consequent':
   assumes \<open>consistent A V\<close> \<open>p \<in> V\<close> \<open>tautology (p \<^bold>\<longrightarrow> q)\<close>
   shows \<open>consistent A ({q} \<union> V)\<close>
   using assms consistent_consequent A1 by blast
 
 lemma consistent_disjuncts:
   assumes \<open>consistent A V\<close> \<open>(p \<^bold>\<or> q) \<in> V\<close>
   shows \<open>consistent A ({p} \<union> V) \<or> consistent A ({q} \<union> V)\<close>
 proof (rule ccontr)
   assume \<open>\<not> ?thesis\<close>
   then have \<open>\<not> consistent A ({p} \<union> V)\<close> \<open>\<not> consistent A ({q} \<union> V)\<close>
     by blast+
 
   then obtain S' T' where
     S': \<open>set S' \<subseteq> V\<close> \<open>A \<turnstile> imply (p # S') \<^bold>\<bottom>\<close> and
     T': \<open>set T' \<subseteq> V\<close> \<open>A \<turnstile> imply (q # T') \<^bold>\<bottom>\<close>
     using \<open>consistent A V\<close> inconsistent_subset by metis
 
   from S' have p: \<open>A \<turnstile> imply (p # S' @ T') \<^bold>\<bottom>\<close>
     by (metis K_imply_weaken Un_upper1 append_Cons set_append)
   moreover from T' have q: \<open>A \<turnstile> imply (q # S' @ T') \<^bold>\<bottom>\<close>
     by (metis K_imply_head K_right_mp R1 imply.simps(2) imply_append)
   ultimately have \<open>A \<turnstile> imply ((p \<^bold>\<or> q) # S' @ T') \<^bold>\<bottom>\<close>
     using K_DisL by blast
   then have \<open>A \<turnstile> imply (S' @ T') \<^bold>\<bottom>\<close>
     using S'(1) T'(1) p q \<open>consistent A V\<close> \<open>(p \<^bold>\<or> q) \<in> V\<close> unfolding consistent_def
     by (metis Un_subset_iff insert_subset list.simps(15) set_append)
   moreover have \<open>set (S' @ T') \<subseteq> V\<close>
     by (simp add: S'(1) T'(1))
   ultimately show False
     using \<open>consistent A V\<close> unfolding consistent_def by blast
 qed
 
 lemma exists_finite_inconsistent:
   assumes \<open>\<not> consistent A ({\<^bold>\<not> p} \<union> V)\<close>
   obtains W where \<open>{\<^bold>\<not> p} \<union> W \<subseteq> {\<^bold>\<not> p} \<union> V\<close> \<open>(\<^bold>\<not> p) \<notin> W\<close> \<open>finite W\<close> \<open>\<not> consistent A ({\<^bold>\<not> p} \<union> W)\<close>
 proof -
   obtain W' where W': \<open>set W' \<subseteq> {\<^bold>\<not> p} \<union> V\<close> \<open>A \<turnstile> imply W' \<^bold>\<bottom>\<close>
     using assms unfolding consistent_def by blast
   let ?S = \<open>removeAll (\<^bold>\<not> p) W'\<close>
   have \<open>\<not> consistent A ({\<^bold>\<not> p} \<union> set ?S)\<close>
     unfolding consistent_def using W'(2) by auto
   moreover have \<open>finite (set ?S)\<close>
     by blast
   moreover have \<open>{\<^bold>\<not> p} \<union> set ?S \<subseteq> {\<^bold>\<not> p} \<union> V\<close>
     using W'(1) by auto
   moreover have \<open>(\<^bold>\<not> p) \<notin> set ?S\<close>
     by simp
   ultimately show ?thesis
     by (meson that)
 qed
 
 lemma inconsistent_imply:
   assumes \<open>\<not> consistent A ({\<^bold>\<not> p} \<union> set G)\<close>
   shows \<open>A \<turnstile> imply G p\<close>
   using assms K_Boole K_imply_weaken unfolding consistent_def
   by (metis insert_is_Un list.simps(15))
 
-lemma consistent_axioms:
-  assumes \<open>\<And>(M :: ('i :: countable, 'w) kripke) w p. A p \<Longrightarrow> P M \<Longrightarrow> M, w \<Turnstile> p\<close> \<open>\<exists>M. P M\<close>
-  shows \<open>consistent A {}\<close>
-proof (rule ccontr)
-  assume \<open>\<not> ?thesis\<close>
-  then have \<open>A \<turnstile> \<^bold>\<bottom>\<close>
-    unfolding consistent_def by simp
-  then have \<open>M, w \<Turnstile> \<^bold>\<bottom>\<close> for M :: \<open>('i, 'w) kripke\<close> and w
-    using soundness assms by (metis semantics.simps(1))
-  then show False
-    by simp
-qed
-
 subsection \<open>Maximal consistent sets\<close>
 
 definition maximal :: \<open>('i fm \<Rightarrow> bool) \<Rightarrow> 'i fm set \<Rightarrow> bool\<close> where
   \<open>maximal A S \<equiv> \<forall>p. p \<notin> S \<longrightarrow> \<not> consistent A ({p} \<union> S)\<close>
 
 theorem deriv_in_maximal:
   assumes \<open>consistent A V\<close> \<open>maximal A V\<close> \<open>A \<turnstile> p\<close>
   shows \<open>p \<in> V\<close>
   using assms R1 inconsistent_subset unfolding consistent_def maximal_def
   by (metis imply.simps(2))
 
 theorem exactly_one_in_maximal:
   assumes \<open>consistent A V\<close> \<open>maximal A V\<close>
   shows \<open>p \<in> V \<longleftrightarrow> (\<^bold>\<not> p) \<notin> V\<close>
 proof
   assume \<open>p \<in> V\<close>
   then show \<open>(\<^bold>\<not> p) \<notin> V\<close>
     using assms K_mp unfolding consistent_def maximal_def
     by (metis empty_subsetI insert_subset list.set(1) list.simps(15))
 next
   assume \<open>(\<^bold>\<not> p) \<notin> V\<close>
   have \<open>A \<turnstile> (p \<^bold>\<or> \<^bold>\<not> p)\<close>
     by (simp add: A1)
   then have \<open>(p \<^bold>\<or> \<^bold>\<not> p) \<in> V\<close>
     using assms deriv_in_maximal by blast
   then have \<open>consistent A ({p} \<union> V) \<or> consistent A ({\<^bold>\<not> p} \<union> V)\<close>
     using assms consistent_disjuncts by blast
   then show \<open>p \<in> V\<close>
     using \<open>maximal A V\<close> \<open>(\<^bold>\<not> p) \<notin> V\<close> unfolding maximal_def by blast
 qed
 
 theorem consequent_in_maximal:
   assumes \<open>consistent A V\<close> \<open>maximal A V\<close> \<open>p \<in> V\<close> \<open>(p \<^bold>\<longrightarrow> q) \<in> V\<close>
   shows \<open>q \<in> V\<close>
 proof -
   have \<open>\<forall>V'. set V' \<subseteq> V \<longrightarrow> \<not> A \<turnstile> imply (p # (p \<^bold>\<longrightarrow> q) # V') \<^bold>\<bottom>\<close>
     using \<open>consistent A V\<close> \<open>p \<in> V\<close> \<open>(p \<^bold>\<longrightarrow> q) \<in> V\<close> unfolding consistent_def
     by (metis insert_subset list.simps(15))
   then have \<open>\<forall>V'. set V' \<subseteq> V \<longrightarrow> \<not> A \<turnstile> imply (q # V') \<^bold>\<bottom>\<close>
     by (meson K_mp K_ImpI K_imply_weaken K_right_mp set_subset_Cons)
   then have \<open>consistent A ({q} \<union> V)\<close>
     using \<open>consistent A V\<close> inconsistent_subset by metis
   then show ?thesis
     using \<open>maximal A V\<close> unfolding maximal_def by fast
 qed
 
 theorem ax_in_maximal:
   assumes \<open>consistent A V\<close> \<open>maximal A V\<close> \<open>A p\<close>
   shows \<open>p \<in> V\<close>
   using assms deriv_in_maximal Ax by blast
 
 theorem mcs_properties:
-  assumes \<open>consistent A V\<close> \<open>maximal A V\<close>
+  assumes \<open>consistent A V\<close> and \<open>maximal A V\<close>
   shows \<open>A \<turnstile> p \<Longrightarrow> p \<in> V\<close>
     and \<open>p \<in> V \<longleftrightarrow> (\<^bold>\<not> p) \<notin> V\<close>
     and \<open>p \<in> V \<Longrightarrow> (p \<^bold>\<longrightarrow> q) \<in> V \<Longrightarrow> q \<in> V\<close>
   using assms deriv_in_maximal exactly_one_in_maximal consequent_in_maximal by blast+
 
 subsection \<open>Lindenbaum extension\<close>
 
 instantiation fm :: (countable) countable begin
 instance by countable_datatype
 end
 
 primrec extend :: \<open>('i fm \<Rightarrow> bool) \<Rightarrow> 'i fm set \<Rightarrow> (nat \<Rightarrow> 'i fm) \<Rightarrow> nat \<Rightarrow> 'i fm set\<close> where
   \<open>extend A S f 0 = S\<close> |
   \<open>extend A S f (Suc n) =
     (if consistent A ({f n} \<union> extend A S f n)
      then {f n} \<union> extend A S f n
      else extend A S f n)\<close>
 
 definition Extend :: \<open>('i fm \<Rightarrow> bool) \<Rightarrow> 'i fm set \<Rightarrow> (nat \<Rightarrow> 'i fm) \<Rightarrow> 'i fm set\<close> where
   \<open>Extend A S f \<equiv> \<Union>n. extend A S f n\<close>
 
 lemma Extend_subset: \<open>S \<subseteq> Extend A S f\<close>
   unfolding Extend_def using Union_upper extend.simps(1) range_eqI
   by metis
 
 lemma extend_bound: \<open>(\<Union>n \<le> m. extend A S f n) = extend A S f m\<close>
   by (induct m) (simp_all add: atMost_Suc)
 
 lemma consistent_extend: \<open>consistent A S \<Longrightarrow> consistent A (extend A S f n)\<close>
   by (induct n) simp_all
 
 lemma UN_finite_bound:
   assumes \<open>finite A\<close> \<open>A \<subseteq> (\<Union>n. f n)\<close>
   shows \<open>\<exists>m :: nat. A \<subseteq> (\<Union>n \<le> m. f n)\<close>
   using assms
 proof (induct rule: finite_induct)
   case (insert x A)
   then obtain m where \<open>A \<subseteq> (\<Union>n \<le> m. f n)\<close>
     by fast
   then have \<open>A \<subseteq> (\<Union>n \<le> (m + k). f n)\<close> for k
     by fastforce
   moreover obtain m' where \<open>x \<in> f m'\<close>
     using insert(4) by blast
   ultimately have \<open>{x} \<union> A \<subseteq> (\<Union>n \<le> m + m'. f n)\<close>
     by auto
   then show ?case
     by blast
 qed simp
 
 lemma consistent_Extend:
   assumes \<open>consistent A S\<close>
   shows \<open>consistent A (Extend A S f)\<close>
   unfolding Extend_def
 proof (rule ccontr)
   assume \<open>\<not> consistent A (\<Union>n. extend A S f n)\<close>
   then obtain S' where \<open>A \<turnstile> imply S' \<^bold>\<bottom>\<close> \<open>set S' \<subseteq> (\<Union>n. extend A S f n)\<close>
     unfolding consistent_def by blast
   then obtain m where \<open>set S' \<subseteq> (\<Union>n \<le> m. extend A S f n)\<close>
     using UN_finite_bound by (metis List.finite_set)
   then have \<open>set S' \<subseteq> extend A S f m\<close>
     using extend_bound by blast
   moreover have \<open>consistent A (extend A S f m)\<close>
     using assms consistent_extend by blast
   ultimately show False
     unfolding consistent_def using \<open>A \<turnstile> imply S' \<^bold>\<bottom>\<close> by blast
 qed
 
 lemma maximal_Extend:
   assumes \<open>surj f\<close>
   shows \<open>maximal A (Extend A S f)\<close>
 proof (rule ccontr)
   assume \<open>\<not> maximal A (Extend A S f)\<close>
   then obtain p where \<open>p \<notin> Extend A S f\<close> \<open>consistent A ({p} \<union> Extend A S f)\<close>
     unfolding maximal_def using assms consistent_Extend by blast
   obtain k where n: \<open>f k = p\<close>
     using \<open>surj f\<close> unfolding surj_def by metis
   then have \<open>p \<notin> extend A S f (Suc k)\<close>
     using \<open>p \<notin> Extend A S f\<close> unfolding Extend_def by blast
   then have \<open>\<not> consistent A ({p} \<union> extend A S f k)\<close>
     using n by fastforce
   moreover have \<open>{p} \<union> extend A S f k \<subseteq> {p} \<union> Extend A S f\<close>
     unfolding Extend_def by blast
   ultimately have \<open>\<not> consistent A ({p} \<union> Extend A S f)\<close>
     unfolding consistent_def by fastforce
   then show False
     using \<open>consistent A ({p} \<union> Extend A S f)\<close> by blast
 qed
 
 lemma maximal_extension:
   fixes V :: \<open>('i :: countable) fm set\<close>
   assumes \<open>consistent A V\<close>
   obtains W where \<open>V \<subseteq> W\<close> \<open>consistent A W\<close> \<open>maximal A W\<close>
 proof -
   let ?W = \<open>Extend A V from_nat\<close>
   have \<open>V \<subseteq> ?W\<close>
     using Extend_subset by blast
   moreover have \<open>consistent A ?W\<close>
     using assms consistent_Extend by blast
   moreover have \<open>maximal A ?W\<close>
     using assms maximal_Extend surj_from_nat by blast
   ultimately show ?thesis
     using that by blast
 qed
 
-lemma ex_mcs:
-  assumes \<open>\<And>(M :: ('i :: countable, 'w) kripke) w p. A p \<Longrightarrow> P M \<Longrightarrow> M, w \<Turnstile> p\<close> \<open>\<exists>M. P M\<close>
-  shows \<open>\<exists>x. x \<in> {V. consistent A V \<and> maximal A V}\<close>
-proof
-  let ?V = \<open>Extend A {\<^bold>\<top>} from_nat\<close>
-  have \<open>consistent A {}\<close>
-    using assms consistent_axioms by blast
-  moreover have \<open>A \<turnstile> \<^bold>\<top>\<close>
-    by (simp add: A1)
-  ultimately have \<open>consistent A {\<^bold>\<top>}\<close>
-    using consistent_deriv by auto
-  then have \<open>consistent A ?V\<close>
-    using consistent_Extend by blast
-  moreover have \<open>maximal A ?V\<close>
-    using maximal_Extend surj_from_nat by blast
-  ultimately show \<open>?V \<in> {V. consistent A V \<and> maximal A V}\<close>
-    by blast
-qed
-
 subsection \<open>Canonical model\<close>
 
 abbreviation pi :: \<open>'i fm set \<Rightarrow> id \<Rightarrow> bool\<close> where
   \<open>pi V x \<equiv> Pro x \<in> V\<close>
 
 abbreviation known :: \<open>'i fm set \<Rightarrow> 'i \<Rightarrow> 'i fm set\<close> where
   \<open>known V i \<equiv> {p. K i p \<in> V}\<close>
 
 abbreviation reach :: \<open>('i fm \<Rightarrow> bool) \<Rightarrow> 'i \<Rightarrow> 'i fm set \<Rightarrow> 'i fm set set\<close> where
-  \<open>reach A i V \<equiv> {W. known V i \<subseteq> W \<and> consistent A W \<and> maximal A W}\<close>
+  \<open>reach A i V \<equiv> {W. known V i \<subseteq> W}\<close>
+
+abbreviation mcss :: \<open>('i fm \<Rightarrow> bool) \<Rightarrow> 'i fm set set\<close> where
+  \<open>mcss A \<equiv> {W. consistent A W \<and> maximal A W}\<close>
 
 lemma truth_lemma:
   fixes A and p :: \<open>('i :: countable) fm\<close>
-  defines \<open>M \<equiv> Kripke pi (reach A)\<close>
-  assumes \<open>consistent A V\<close> \<open>maximal A V\<close>
+  defines \<open>M \<equiv> Kripke (mcss A) pi (reach A)\<close>
+  assumes \<open>consistent A V\<close> and \<open>maximal A V\<close>
   shows \<open>(p \<in> V \<longleftrightarrow> M, V \<Turnstile> p) \<and> ((\<^bold>\<not> p) \<in> V \<longleftrightarrow> M, V \<Turnstile> \<^bold>\<not> p)\<close>
   using assms unfolding M_def
 proof (induct p arbitrary: V)
   case FF
   then show ?case
   proof (intro conjI impI iffI)
     assume \<open>\<^bold>\<bottom> \<in> V\<close>
     then have False
       using \<open>consistent A V\<close> K_imply_head unfolding consistent_def
       by (metis bot.extremum insert_subset list.set(1) list.simps(15))
-    then show \<open>Kripke pi (reach A), V \<Turnstile> \<^bold>\<bottom>\<close> ..
+    then show \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<bottom>\<close> ..
   next
-    assume \<open>Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> \<^bold>\<bottom>\<close>
+    assume \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> \<^bold>\<bottom>\<close>
     then show \<open>(\<^bold>\<not> \<^bold>\<bottom>) \<in> V\<close>
       using \<open>consistent A V\<close> \<open>maximal A V\<close> unfolding maximal_def
       by (meson K_Boole inconsistent_subset consistent_def)
   qed simp_all
 next
   case (Pro x)
   then show ?case
   proof (intro conjI impI iffI)
     assume \<open>(\<^bold>\<not> Pro x) \<in> V\<close>
-    then show \<open>Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> Pro x\<close>
+    then show \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> Pro x\<close>
       using \<open>consistent A V\<close> \<open>maximal A V\<close> exactly_one_in_maximal by auto
   next
-    assume \<open>Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> Pro x\<close>
+    assume \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> Pro x\<close>
     then show \<open>(\<^bold>\<not> Pro x) \<in> V\<close>
       using \<open>consistent A V\<close> \<open>maximal A V\<close> exactly_one_in_maximal by auto
   qed (simp_all add: \<open>maximal A V\<close> maximal_def)
 next
   case (Dis p q)
-  have \<open>(p \<^bold>\<or> q) \<in> V \<longrightarrow> Kripke pi (reach A), V \<Turnstile> (p \<^bold>\<or> q)\<close>
+  have \<open>(p \<^bold>\<or> q) \<in> V \<longrightarrow> Kripke (mcss A) pi (reach A), V \<Turnstile> (p \<^bold>\<or> q)\<close>
   proof
     assume \<open>(p \<^bold>\<or> q) \<in> V\<close>
     then have \<open>consistent A ({p} \<union> V) \<or> consistent A ({q} \<union> V)\<close>
       using \<open>consistent A V\<close> consistent_disjuncts by blast
     then have \<open>p \<in> V \<or> q \<in> V\<close>
       using \<open>maximal A V\<close> unfolding maximal_def by fast
-    then show \<open>Kripke pi (reach A), V \<Turnstile> (p \<^bold>\<or> q)\<close>
+    then show \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> (p \<^bold>\<or> q)\<close>
       using Dis by simp
   qed
-  moreover have \<open>(\<^bold>\<not> (p \<^bold>\<or> q)) \<in> V \<longrightarrow> Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<or> q)\<close>
+  moreover have \<open>(\<^bold>\<not> (p \<^bold>\<or> q)) \<in> V \<longrightarrow> Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<or> q)\<close>
   proof
     assume \<open>(\<^bold>\<not> (p \<^bold>\<or> q)) \<in> V\<close>
     then have \<open>consistent A ({\<^bold>\<not> q} \<union> V)\<close> \<open>consistent A ({\<^bold>\<not> p} \<union> V)\<close>
       using \<open>consistent A V\<close> consistent_consequent' by fastforce+
     then have \<open>(\<^bold>\<not> p) \<in> V\<close> \<open>(\<^bold>\<not> q) \<in> V\<close>
       using \<open>maximal A V\<close> unfolding maximal_def by fast+
-    then show \<open>Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<or> q)\<close>
+    then show \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<or> q)\<close>
       using Dis by simp
   qed
   ultimately show ?case
     using exactly_one_in_maximal Dis by auto
 next
   case (Con p q)
-  have \<open>(p \<^bold>\<and> q) \<in> V \<longrightarrow> Kripke pi (reach A), V \<Turnstile> (p \<^bold>\<and> q)\<close>
+  have \<open>(p \<^bold>\<and> q) \<in> V \<longrightarrow> Kripke (mcss A) pi (reach A), V \<Turnstile> (p \<^bold>\<and> q)\<close>
   proof
     assume \<open>(p \<^bold>\<and> q) \<in> V\<close>
     then have \<open>consistent A ({p} \<union> V)\<close> \<open>consistent A ({q} \<union> V)\<close>
       using \<open>consistent A V\<close> consistent_consequent' by fastforce+
     then have \<open>p \<in> V\<close> \<open>q \<in> V\<close>
       using \<open>maximal A V\<close> unfolding maximal_def by fast+
-    then show \<open>Kripke pi (reach A), V \<Turnstile> (p \<^bold>\<and> q)\<close>
+    then show \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> (p \<^bold>\<and> q)\<close>
       using Con by simp
   qed
-  moreover have \<open>(\<^bold>\<not> (p \<^bold>\<and> q)) \<in> V \<longrightarrow> Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<and> q)\<close>
+  moreover have \<open>(\<^bold>\<not> (p \<^bold>\<and> q)) \<in> V \<longrightarrow> Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<and> q)\<close>
   proof
     assume \<open>(\<^bold>\<not> (p \<^bold>\<and> q)) \<in> V\<close>
     then have \<open>consistent A ({\<^bold>\<not> p \<^bold>\<or> \<^bold>\<not> q} \<union> V)\<close>
       using \<open>consistent A V\<close> consistent_consequent' by fastforce
     then have \<open>consistent A ({\<^bold>\<not> p} \<union> V) \<or> consistent A ({\<^bold>\<not> q} \<union> V)\<close>
       using \<open>consistent A V\<close> \<open>maximal A V\<close> consistent_disjuncts unfolding maximal_def by blast
     then have \<open>(\<^bold>\<not> p) \<in> V \<or> (\<^bold>\<not> q) \<in> V\<close>
       using \<open>maximal A V\<close> unfolding maximal_def by fast
-    then show \<open>Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<and> q)\<close>
+    then show \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<and> q)\<close>
       using Con by simp
   qed
   ultimately show ?case
     using exactly_one_in_maximal Con by auto
 next
   case (Imp p q)
-  have \<open>(p \<^bold>\<longrightarrow> q) \<in> V \<longrightarrow> Kripke pi (reach A), V \<Turnstile> (p \<^bold>\<longrightarrow> q)\<close>
+  have \<open>(p \<^bold>\<longrightarrow> q) \<in> V \<longrightarrow> Kripke (mcss A) pi (reach A), V \<Turnstile> (p \<^bold>\<longrightarrow> q)\<close>
   proof
     assume \<open>(p \<^bold>\<longrightarrow> q) \<in> V\<close>
     then have \<open>consistent A ({\<^bold>\<not> p \<^bold>\<or> q} \<union> V)\<close>
       using \<open>consistent A V\<close> consistent_consequent' by fastforce
     then have \<open>consistent A ({\<^bold>\<not> p} \<union> V) \<or> consistent A ({q} \<union> V)\<close>
       using \<open>consistent A V\<close> \<open>maximal A V\<close> consistent_disjuncts unfolding maximal_def by blast
     then have \<open>(\<^bold>\<not> p) \<in> V \<or> q \<in> V\<close>
       using \<open>maximal A V\<close> unfolding maximal_def by fast
-    then show \<open>Kripke pi (reach A), V \<Turnstile> (p \<^bold>\<longrightarrow> q)\<close>
+    then show \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> (p \<^bold>\<longrightarrow> q)\<close>
       using Imp by simp
   qed
-  moreover have \<open>(\<^bold>\<not> (p \<^bold>\<longrightarrow> q)) \<in> V \<longrightarrow> Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<longrightarrow> q)\<close>
+  moreover have \<open>(\<^bold>\<not> (p \<^bold>\<longrightarrow> q)) \<in> V \<longrightarrow> Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<longrightarrow> q)\<close>
   proof
     assume \<open>(\<^bold>\<not> (p \<^bold>\<longrightarrow> q)) \<in> V\<close>
     then have \<open>consistent A ({p} \<union> V)\<close> \<open>consistent A ({\<^bold>\<not> q} \<union> V)\<close>
       using \<open>consistent A V\<close> consistent_consequent' by fastforce+
     then have \<open>p \<in> V\<close> \<open>(\<^bold>\<not> q) \<in> V\<close>
       using \<open>maximal A V\<close> unfolding maximal_def by fast+
-    then show \<open>Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<longrightarrow> q)\<close>
+    then show \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> (p \<^bold>\<longrightarrow> q)\<close>
       using Imp by simp
   qed
   ultimately show ?case
     using exactly_one_in_maximal Imp \<open>consistent A V\<close> by auto
 next
   case (K i p)
-  then have \<open>K i p \<in> V \<longrightarrow> Kripke pi (reach A), V \<Turnstile> K i p\<close>
+  then have \<open>K i p \<in> V \<longrightarrow> Kripke (mcss A) pi (reach A), V \<Turnstile> K i p\<close>
     by auto
-  moreover have \<open>(Kripke pi (reach A), V \<Turnstile> K i p) \<longrightarrow> K i p \<in> V\<close>
+  moreover have \<open>(Kripke (mcss A) pi (reach A), V \<Turnstile> K i p) \<longrightarrow> K i p \<in> V\<close>
   proof (intro allI impI)
-    assume \<open>Kripke pi (reach A), V \<Turnstile> K i p\<close>
+    assume \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> K i p\<close>
 
     have \<open>\<not> consistent A ({\<^bold>\<not> p} \<union> known V i)\<close>
     proof
       assume \<open>consistent A ({\<^bold>\<not> p} \<union> known V i)\<close>
       then obtain W where W: \<open>{\<^bold>\<not> p} \<union> known V i \<subseteq> W\<close> \<open>consistent A W\<close> \<open>maximal A W\<close>
         using \<open>consistent A V\<close> maximal_extension by blast
-      then have \<open>Kripke pi (reach A), W \<Turnstile> \<^bold>\<not> p\<close>
+      then have \<open>Kripke (mcss A) pi (reach A), W \<Turnstile> \<^bold>\<not> p\<close>
         using K \<open>consistent A V\<close> by blast
-      moreover have \<open>W \<in> reach A i V\<close>
-        using W by simp
-      ultimately have \<open>Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> K i p\<close>
+      moreover have \<open>W \<in> reach A i V\<close> \<open>W \<in> mcss A\<close>
+        using W by simp_all
+      ultimately have \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> K i p\<close>
         by auto
       then show False
-        using \<open>Kripke pi (reach A), V \<Turnstile> K i p\<close> by auto
+        using \<open>Kripke (mcss A) pi (reach A), V \<Turnstile> K i p\<close> by auto
     qed
 
     then obtain W where W:
       \<open>{\<^bold>\<not> p} \<union> W \<subseteq> {\<^bold>\<not> p} \<union> known V i\<close> \<open>(\<^bold>\<not> p) \<notin> W\<close> \<open>finite W\<close> \<open>\<not> consistent A ({\<^bold>\<not> p} \<union> W)\<close>
       using exists_finite_inconsistent by metis
 
     obtain L where L: \<open>set L = W\<close>
       using \<open>finite W\<close> finite_list by blast
 
     then have \<open>A \<turnstile> imply L p\<close>
       using W(4) inconsistent_imply by blast
     then have \<open>A \<turnstile> K i (imply L p)\<close>
       using R2 by fast
     then have \<open>A \<turnstile> imply (map (K i) L) (K i p)\<close>
       using K_distrib_K_imp by fast
     then have \<open>imply (map (K i) L) (K i p) \<in> V\<close>
       using deriv_in_maximal K.prems(1, 2) by blast
     then show \<open>K i p \<in> V\<close>
       using L W(1-2)
     proof (induct L arbitrary: W)
       case (Cons a L)
       then have \<open>K i a \<in> V\<close>
         by auto
       then have \<open>imply (map (K i) L) (K i p) \<in> V\<close>
         using Cons(2) \<open>consistent A V\<close> \<open>maximal A V\<close> consequent_in_maximal by auto
       then show ?case
         using Cons by auto
     qed simp
   qed
-  moreover have \<open>(Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> K i p) \<longrightarrow> (\<^bold>\<not> K i p) \<in> V\<close>
+  moreover have \<open>(Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> K i p) \<longrightarrow> (\<^bold>\<not> K i p) \<in> V\<close>
     using \<open>consistent A V\<close> \<open>maximal A V\<close> exactly_one_in_maximal calculation(1)
     by (metis (no_types, lifting) semantics.simps(1, 5))
-  moreover have \<open>(\<^bold>\<not> K i p) \<in> V \<longrightarrow> Kripke pi (reach A), V \<Turnstile> \<^bold>\<not> K i p\<close>
+  moreover have \<open>(\<^bold>\<not> K i p) \<in> V \<longrightarrow> Kripke (mcss A) pi (reach A), V \<Turnstile> \<^bold>\<not> K i p\<close>
     using \<open>consistent A V\<close> \<open>maximal A V\<close> calculation(2) exactly_one_in_maximal by auto
   ultimately show ?case
     by blast
 qed
 
 lemma canonical_model:
-  assumes \<open>consistent A S\<close> \<open>p \<in> S\<close>
-  defines \<open>V \<equiv> Extend A S from_nat\<close> and \<open>M \<equiv> Kripke pi (reach A)\<close>
-  shows \<open>M, V \<Turnstile> p\<close> \<open>consistent A V\<close> \<open>maximal A V\<close>
+  assumes \<open>consistent A S\<close> and \<open>p \<in> S\<close>
+  defines \<open>V \<equiv> Extend A S from_nat\<close> and \<open>M \<equiv> Kripke (mcss A) pi (reach A)\<close>
+  shows \<open>M, V \<Turnstile> p\<close> and \<open>consistent A V\<close> and \<open>maximal A V\<close>
 proof -
   have \<open>consistent A V\<close>
     using \<open>consistent A S\<close> unfolding V_def using consistent_Extend by blast
   have \<open>maximal A V\<close>
     unfolding V_def using maximal_Extend surj_from_nat by blast
 
   { fix x
     assume \<open>x \<in> S\<close>
     then have \<open>x \<in> V\<close>
       unfolding V_def using Extend_subset by blast
     then have \<open>M, V \<Turnstile> x\<close>
       unfolding M_def using truth_lemma \<open>consistent A V\<close> \<open>maximal A V\<close> by blast }
   then show \<open>M, V \<Turnstile> p\<close>
     using \<open>p \<in> S\<close> by blast+
   show \<open>consistent A V\<close> \<open>maximal A V\<close>
     by fact+
 qed
 
 subsection \<open>Completeness\<close>
 
 lemma imply_completeness:
-  assumes valid: \<open>\<forall>(M :: ('i, ('i :: countable) fm set) kripke) w.
-    list_all (\<lambda>q. M, w \<Turnstile> q) G \<longrightarrow> M, w \<Turnstile> p\<close>
-  shows \<open>A \<turnstile> imply G p\<close>
-proof (rule K_Boole, rule ccontr)
-  assume \<open>\<not> A \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close>
+  assumes valid: \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M.
+    (\<forall>q \<in> G. M, w \<Turnstile> q) \<longrightarrow> M, w \<Turnstile> p\<close>
+  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (A \<turnstile> imply qs p)\<close>
+proof (rule ccontr)
+  assume \<open>\<nexists>qs. set qs \<subseteq> G \<and> A \<turnstile> imply qs p\<close>
+  then have *: \<open>\<forall>qs. set qs \<subseteq> G \<longrightarrow> \<not> A \<turnstile> imply ((\<^bold>\<not> p) # qs) \<^bold>\<bottom>\<close>
+    using K_Boole by blast
 
-  let ?S = \<open>set ((\<^bold>\<not> p) # G)\<close>
+  let ?S = \<open>{\<^bold>\<not> p} \<union> G\<close>
   let ?V = \<open>Extend A ?S from_nat\<close>
-  let ?M = \<open>Kripke pi (reach A)\<close>
+  let ?M = \<open>Kripke (mcss A) pi (reach A)\<close>
 
   have \<open>consistent A ?S\<close>
-    unfolding consistent_def using \<open>\<not> A \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close> K_imply_weaken by blast
-  then have \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>list_all (\<lambda>p. ?M, ?V \<Turnstile> p) G\<close>
-    unfolding list_all_def using canonical_model by fastforce+
-  then have \<open>?M, ?V \<Turnstile> p\<close>
-    using valid by blast
+    using * by (metis K_imply_Cons consistent_def inconsistent_subset)
+  then have \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>\<forall>q \<in> G. ?M, ?V \<Turnstile> q\<close>
+    using canonical_model by fastforce+
+  moreover have \<open>?V \<in> mcss A\<close>
+    using \<open>consistent A ?S\<close> consistent_Extend maximal_Extend surj_from_nat by blast
+  ultimately have \<open>?M, ?V \<Turnstile> p\<close>
+    using valid by simp
   then show False
     using \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> by simp
 qed
 
 theorem completeness:
-  assumes \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke) w. M, w \<Turnstile> p\<close>
+  assumes \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M. M, w \<Turnstile> p\<close>
   shows \<open>A \<turnstile> p\<close>
-  using assms imply_completeness[where G=\<open>[]\<close>] by auto
+  using assms imply_completeness[where G=\<open>{}\<close>] by auto
 
 section \<open>System K\<close>
 
 abbreviation SystemK :: \<open>'i fm \<Rightarrow> bool\<close> ("\<turnstile>\<^sub>K _" [50] 50) where
   \<open>\<turnstile>\<^sub>K p \<equiv> (\<lambda>_. False) \<turnstile> p\<close>
 
-lemma soundness\<^sub>K: \<open>\<turnstile>\<^sub>K p \<Longrightarrow> M, w \<Turnstile> p\<close>
+lemma soundness\<^sub>K: \<open>\<turnstile>\<^sub>K p \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
   using soundness by metis
 
-abbreviation \<open>valid\<^sub>K p \<equiv> \<forall>(M :: (nat, nat fm set) kripke) w. M, w \<Turnstile> p\<close>
+abbreviation \<open>valid\<^sub>K p \<equiv> \<forall>(M :: (nat, nat fm set) kripke). \<forall>w \<in> \<W> M. M, w \<Turnstile> p\<close>
 
 theorem main\<^sub>K: \<open>valid\<^sub>K p \<longleftrightarrow> \<turnstile>\<^sub>K p\<close>
 proof
   assume \<open>valid\<^sub>K p\<close>
   with completeness show \<open>\<turnstile>\<^sub>K p\<close>
     by blast
 next
   assume \<open>\<turnstile>\<^sub>K p\<close>
   with soundness\<^sub>K show \<open>valid\<^sub>K p\<close>
-    by (intro allI)
+    by fast
 qed
 
-corollary \<open>valid\<^sub>K p \<longrightarrow> M, w \<Turnstile> p\<close>
-proof
-  assume \<open>valid\<^sub>K p\<close>
-  then have \<open>\<turnstile>\<^sub>K p\<close>
-    unfolding main\<^sub>K .
-  with soundness\<^sub>K show \<open>M, w \<Turnstile> p\<close> .
+corollary
+  assumes \<open>valid\<^sub>K p\<close> and \<open>w \<in> \<W> M\<close>
+  shows \<open>M, w \<Turnstile> p\<close>
+proof -
+  have \<open>\<turnstile>\<^sub>K p\<close>
+    using assms(1) unfolding main\<^sub>K .
+  with soundness\<^sub>K assms(2) show \<open>M, w \<Turnstile> p\<close> by fast
 qed
 
 section \<open>System T\<close>
 
 text \<open>Also known as System M\<close>
 
 inductive AxT :: \<open>'i fm \<Rightarrow> bool\<close> where
   \<open>AxT (K i p \<^bold>\<longrightarrow> p)\<close>
 
 abbreviation SystemT :: \<open>'i fm \<Rightarrow> bool\<close> ("\<turnstile>\<^sub>T _" [50] 50) where
   \<open>\<turnstile>\<^sub>T p \<equiv> AxT \<turnstile> p\<close>
 
-lemma soundness_AxT: \<open>AxT p \<Longrightarrow> reflexive M \<Longrightarrow> M, w \<Turnstile> p\<close>
+lemma soundness_AxT: \<open>AxT p \<Longrightarrow> reflexive M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
   by (induct p rule: AxT.induct) (meson truth)
 
-lemma soundness\<^sub>T: \<open>\<turnstile>\<^sub>T p \<Longrightarrow> reflexive M \<Longrightarrow> M, w \<Turnstile> p\<close>
+lemma soundness\<^sub>T: \<open>\<turnstile>\<^sub>T p \<Longrightarrow> reflexive M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
   using soundness soundness_AxT .
 
-typedef 'i mcs\<^sub>T = \<open>{V :: ('i :: countable) fm set. consistent AxT V \<and> maximal AxT V}\<close>
-  using ex_mcs[where A=AxT and P=reflexive] soundness_AxT ex_equiv_model by fast
-
-abbreviation \<open>pi\<^sub>T \<equiv> pi o Rep_mcs\<^sub>T\<close>
-abbreviation \<open>reach\<^sub>T i V \<equiv> Abs_mcs\<^sub>T ` (reach AxT i (Rep_mcs\<^sub>T V))\<close>
-
-lemma mcs\<^sub>T_equiv:
-  assumes \<open>consistent AxT V\<close> \<open>maximal AxT V\<close>
-  shows \<open>(Kripke pi (reach AxT), V \<Turnstile> p) = (Kripke pi\<^sub>T reach\<^sub>T, Abs_mcs\<^sub>T V \<Turnstile> p)\<close>
-  using assms by (induct p arbitrary: V) (simp_all add: Abs_mcs\<^sub>T_inverse)
-
 lemma AxT_reflexive:
-  assumes \<open>\<And>p. AxT p \<Longrightarrow> A p\<close> \<open>consistent A V\<close> \<open>maximal A V\<close>
+  assumes \<open>\<forall>p. AxT p \<longrightarrow> A p\<close> and \<open>consistent A V\<close> and \<open>maximal A V\<close>
   shows \<open>V \<in> reach A i V\<close>
 proof -
   have \<open>(K i p \<^bold>\<longrightarrow> p) \<in> V\<close> for p
     using assms ax_in_maximal AxT.intros by metis
   then have \<open>p \<in> V\<close> if \<open>K i p \<in> V\<close> for p
     using that assms consequent_in_maximal by blast
   then show ?thesis
     using assms by blast
 qed
 
-lemma mcs\<^sub>T_reflexive: \<open>reflexive (Kripke pi\<^sub>T reach\<^sub>T)\<close>
+lemma mcs\<^sub>T_reflexive:
+  assumes \<open>\<forall>p. AxT p \<longrightarrow> A p\<close>
+  shows \<open>reflexive (Kripke (mcss A) pi (reach A))\<close>
   unfolding reflexive_def
-proof (intro allI)
-  fix i :: \<open>'i :: countable\<close> and V :: \<open>'i mcs\<^sub>T\<close>
-  let ?V = \<open>Rep_mcs\<^sub>T V\<close>
-  have \<open>consistent AxT ?V\<close> \<open>maximal AxT ?V\<close>
-    using Rep_mcs\<^sub>T by blast+
-  then have \<open>?V \<in> reach AxT i ?V\<close>
-    using AxT_reflexive by fast
-  then have \<open>V \<in> reach\<^sub>T i V\<close>
-    using Rep_mcs\<^sub>T_inverse by (metis (no_types, lifting) image_iff)
-  then show \<open>V \<in> \<K> (Kripke pi\<^sub>T reach\<^sub>T) i V\<close>
+proof safe
+  fix i V
+  assume \<open>V \<in> \<W> (Kripke (mcss A) pi (reach A))\<close>
+  then have \<open>consistent A V\<close> \<open>maximal A V\<close>
+    by simp_all
+  with AxT_reflexive assms have \<open>V \<in> reach A i V\<close> .
+  then show \<open>V \<in> \<K> (Kripke (mcss A) pi (reach A)) i V\<close>
     by simp
 qed
 
 lemma imply_completeness_T:
-  assumes valid: \<open>\<forall>(M :: ('i, ('i :: countable) mcs\<^sub>T) kripke) w.
-    reflexive M \<longrightarrow> list_all (\<lambda>q. M, w \<Turnstile> q) G \<longrightarrow> M, w \<Turnstile> p\<close>
-  shows \<open>AxT \<turnstile> imply G p\<close>
-proof (rule K_Boole, rule ccontr)
-  assume \<open>\<not> AxT \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close>
+  assumes valid: \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M.
+    reflexive M \<longrightarrow> (\<forall>q \<in> G. M, w \<Turnstile> q) \<longrightarrow> M, w \<Turnstile> p\<close>
+  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (AxT \<turnstile> imply qs p)\<close>
+proof (rule ccontr)
+  assume \<open>\<nexists>qs. set qs \<subseteq> G \<and> AxT \<turnstile> imply qs p\<close>
+  then have *: \<open>\<forall>qs. set qs \<subseteq> G \<longrightarrow> \<not> AxT \<turnstile> imply ((\<^bold>\<not> p) # qs) \<^bold>\<bottom>\<close>
+    using K_Boole by blast
 
-  let ?S = \<open>set ((\<^bold>\<not> p) # G)\<close>
+  let ?S = \<open>{\<^bold>\<not> p} \<union> G\<close>
   let ?V = \<open>Extend AxT ?S from_nat\<close>
-  let ?M = \<open>Kripke pi (reach AxT)\<close>
-  let ?V\<^sub>T = \<open>Abs_mcs\<^sub>T ?V\<close>
-  let ?M\<^sub>T = \<open>Kripke pi\<^sub>T reach\<^sub>T\<close>
+  let ?M = \<open>Kripke (mcss AxT) pi (reach AxT)\<close>
 
   have \<open>consistent AxT ?S\<close>
-    unfolding consistent_def using \<open>\<not> AxT \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close> K_imply_weaken by blast
-  then have \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>list_all (\<lambda>p. ?M, ?V \<Turnstile> p) G\<close> \<open>consistent AxT ?V\<close> \<open>maximal AxT ?V\<close>
+    using * by (metis K_imply_Cons consistent_def inconsistent_subset)
+  then have \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>\<forall>q \<in> G. ?M, ?V \<Turnstile> q\<close> \<open>consistent AxT ?V\<close> \<open>maximal AxT ?V\<close>
     using canonical_model unfolding list_all_def by fastforce+
-  then have \<open>?M\<^sub>T, ?V\<^sub>T \<Turnstile> (\<^bold>\<not> p)\<close> \<open>list_all (\<lambda>p. ?M\<^sub>T, ?V\<^sub>T \<Turnstile> p) G\<close>
-    using \<open>consistent AxT ?V\<close> \<open>maximal AxT ?V\<close> mcs\<^sub>T_equiv unfolding list_all_def by blast+
-  moreover have \<open>reflexive ?M\<^sub>T\<close>
+  moreover have \<open>reflexive ?M\<close>
     using mcs\<^sub>T_reflexive by fast
-  ultimately have \<open>?M\<^sub>T, ?V\<^sub>T \<Turnstile> p\<close>
-    using valid by blast
+  ultimately have \<open>?M, ?V \<Turnstile> p\<close>
+    using valid by auto
   then show False
-    using \<open>?M\<^sub>T, ?V\<^sub>T \<Turnstile> (\<^bold>\<not> p)\<close> by simp
+    using \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> by simp
 qed
 
 lemma completeness\<^sub>T:
-  assumes \<open>\<forall>(M :: ('i, ('i :: countable) mcs\<^sub>T) kripke) w. reflexive M \<longrightarrow> M, w \<Turnstile> p\<close>
+  assumes \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M. reflexive M \<longrightarrow> M, w \<Turnstile> p\<close>
   shows \<open>\<turnstile>\<^sub>T p\<close>
-  using assms imply_completeness_T[where G=\<open>[]\<close>] by auto
+  using assms imply_completeness_T[where G=\<open>{}\<close>] by auto
 
-abbreviation \<open>valid\<^sub>T p \<equiv> \<forall>(M :: (nat, nat mcs\<^sub>T) kripke) w. reflexive M \<longrightarrow> M, w \<Turnstile> p\<close>
+abbreviation \<open>valid\<^sub>T p \<equiv> \<forall>(M :: (nat, nat fm set) kripke). \<forall>w \<in> \<W> M. reflexive M \<longrightarrow> M, w \<Turnstile> p\<close>
 
 theorem main\<^sub>T: \<open>valid\<^sub>T p \<longleftrightarrow> \<turnstile>\<^sub>T p\<close>
   using soundness\<^sub>T completeness\<^sub>T by fast
 
 corollary
-  assumes \<open>reflexive M\<close>
+  assumes \<open>reflexive M\<close> \<open>w \<in> \<W> M\<close>
   shows \<open>valid\<^sub>T p \<longrightarrow> M, w \<Turnstile> p\<close>
   using assms soundness\<^sub>T completeness\<^sub>T by fast
 
 section \<open>System KB\<close>
 
 inductive AxB :: \<open>'i fm \<Rightarrow> bool\<close> where
   \<open>AxB (p \<^bold>\<longrightarrow> K i (L i p))\<close>
 
 abbreviation SystemKB :: \<open>'i fm \<Rightarrow> bool\<close> ("\<turnstile>\<^sub>K\<^sub>B _" [50] 50) where
   \<open>\<turnstile>\<^sub>K\<^sub>B p \<equiv> AxB \<turnstile> p\<close>
 
-lemma soundness_AxB: \<open>AxB p \<Longrightarrow> symmetric M \<Longrightarrow> M, w \<Turnstile> p\<close>
+lemma soundness_AxB: \<open>AxB p \<Longrightarrow> symmetric M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
   unfolding symmetric_def by (induct p rule: AxB.induct) auto
 
-lemma soundness\<^sub>K\<^sub>B: \<open>\<turnstile>\<^sub>K\<^sub>B p \<Longrightarrow> symmetric M \<Longrightarrow> M, w \<Turnstile> p\<close>
+lemma soundness\<^sub>K\<^sub>B: \<open>\<turnstile>\<^sub>K\<^sub>B p \<Longrightarrow> symmetric M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
   using soundness soundness_AxB .
 
-typedef 'i mcs\<^sub>K\<^sub>B = \<open>{V :: ('i :: countable) fm set. consistent AxB V \<and> maximal AxB V}\<close>
-  using ex_mcs[where A=AxB and P=symmetric] soundness_AxB ex_equiv_model by fast
-
-abbreviation \<open>pi\<^sub>K\<^sub>B \<equiv> pi o Rep_mcs\<^sub>K\<^sub>B\<close>
-abbreviation \<open>reach\<^sub>K\<^sub>B i V \<equiv> Abs_mcs\<^sub>K\<^sub>B ` (reach AxB i (Rep_mcs\<^sub>K\<^sub>B V))\<close>
-
-lemma mcs\<^sub>K\<^sub>B_equiv:
-  assumes \<open>consistent AxB V\<close> \<open>maximal AxB V\<close>
-  shows \<open>(Kripke pi (reach AxB), V \<Turnstile> p) = (Kripke pi\<^sub>K\<^sub>B reach\<^sub>K\<^sub>B, Abs_mcs\<^sub>K\<^sub>B V \<Turnstile> p)\<close>
-  using assms by (induct p arbitrary: V) (simp_all add: Abs_mcs\<^sub>K\<^sub>B_inverse)
-
 lemma AxB_symmetric':
-  assumes \<open>\<And>p. AxB p \<Longrightarrow> A p\<close> \<open>consistent A V\<close> \<open>maximal A V\<close> \<open>consistent A W\<close> \<open>maximal A W\<close>
+  assumes \<open>\<forall>p. AxB p \<longrightarrow> A p\<close> \<open>consistent A V\<close> \<open>maximal A V\<close> \<open>consistent A W\<close> \<open>maximal A W\<close>
     and \<open>W \<in> reach A i V\<close>
   shows \<open>V \<in> reach A i W\<close>
 proof -
   have \<open>\<forall>p. K i p \<in> W \<longrightarrow> p \<in> V\<close>
   proof (intro allI impI, rule ccontr)
     fix p
     assume \<open>K i p \<in> W\<close> \<open>p \<notin> V\<close>
     then have \<open>(\<^bold>\<not> p) \<in> V\<close>
       using assms(2-3) exactly_one_in_maximal by fast
     then have \<open>K i (L i (\<^bold>\<not> p)) \<in> V\<close>
       using assms(1-3) ax_in_maximal AxB.intros consequent_in_maximal by fast
     then have \<open>L i (\<^bold>\<not> p) \<in> W\<close>
       using \<open>W \<in> reach A i V\<close> by fast
     then have \<open>(\<^bold>\<not> K i p) \<in> W\<close>
       using assms(4-5) by (meson K_LK consistent_consequent maximal_def)
     then show False
       using \<open>K i p \<in> W\<close> assms(4-5) exactly_one_in_maximal by fast
   qed
   then have \<open>known W i \<subseteq> V\<close>
     by blast
   then show ?thesis
     using assms(2-3) by simp
 qed
 
 lemma AxB_symmetric:
-  assumes \<open>\<And>p. AxB p \<Longrightarrow> A p\<close> \<open>consistent A V\<close> \<open>maximal A V\<close> \<open>consistent A W\<close> \<open>maximal A W\<close>
+  assumes \<open>\<forall>p. AxB p \<longrightarrow> A p\<close> \<open>consistent A V\<close> \<open>maximal A V\<close> \<open>consistent A W\<close> \<open>maximal A W\<close>
   shows \<open>W \<in> reach A i V \<longleftrightarrow> V \<in> reach A i W\<close>
   using assms AxB_symmetric'[where V=V and W=W] AxB_symmetric'[where V=W and W=V]
   by (intro iffI) blast+
 
-lemma mcs\<^sub>K\<^sub>B_symmetric: \<open>symmetric (Kripke pi\<^sub>K\<^sub>B reach\<^sub>K\<^sub>B)\<close>
+lemma mcs\<^sub>K\<^sub>B_symmetric:
+  assumes \<open>\<forall>p. AxB p \<longrightarrow> A p\<close>
+  shows \<open>symmetric (Kripke (mcss A) pi (reach A))\<close>
   unfolding symmetric_def
-proof (intro allI)
-  fix i :: \<open>'i :: countable\<close> and V W :: \<open>'i mcs\<^sub>K\<^sub>B\<close>
-  let ?V = \<open>Rep_mcs\<^sub>K\<^sub>B V\<close> and ?W = \<open>Rep_mcs\<^sub>K\<^sub>B W\<close>
-
-  have \<open>consistent AxB ?V\<close> \<open>maximal AxB ?V\<close>
-    \<open>consistent AxB ?W\<close> \<open>maximal AxB ?W\<close>
-    using Rep_mcs\<^sub>K\<^sub>B by blast+
-  then have \<open>?V \<in> reach AxB i ?W \<longleftrightarrow> ?W \<in> reach AxB i ?V\<close>
-    using AxB_symmetric' by fast
-  then have \<open>V \<in> reach\<^sub>K\<^sub>B i W \<longleftrightarrow> W \<in> reach\<^sub>K\<^sub>B i V\<close>
-    by (intro iffI)
-      (metis (no_types, lifting) Abs_mcs\<^sub>K\<^sub>B_inverse Rep_mcs\<^sub>K\<^sub>B_inverse image_iff mem_Collect_eq)+
-  then show \<open>(W \<in> \<K> (Kripke pi\<^sub>K\<^sub>B reach\<^sub>K\<^sub>B) i V) = (V \<in> \<K> (Kripke pi\<^sub>K\<^sub>B reach\<^sub>K\<^sub>B) i W)\<close>
+proof (intro allI ballI)
+  fix i V W
+  assume \<open>V \<in> \<W> (Kripke (mcss A) pi (reach A))\<close> \<open>W \<in> \<W> (Kripke (mcss A) pi (reach A))\<close>
+  then have \<open>consistent A V\<close> \<open>maximal A V\<close> \<open>consistent A W\<close> \<open>maximal A W\<close>
+    by simp_all
+  with AxB_symmetric assms have \<open>W \<in> reach A i V \<longleftrightarrow> V \<in> reach A i W\<close> .
+  then show
+    \<open>(W \<in> \<K> (Kripke (mcss A) pi (reach A)) i V) = (V \<in> \<K> (Kripke (mcss A) pi (reach A)) i W)\<close>
     by simp
 qed
 
 lemma imply_completeness_KB:
-  assumes valid: \<open>\<forall>(M :: ('i :: countable, 'i mcs\<^sub>K\<^sub>B) kripke) w.
-    symmetric M \<longrightarrow> list_all (\<lambda>q. M, w \<Turnstile> q) G \<longrightarrow> M, w \<Turnstile> p\<close>
-  shows \<open>AxB \<turnstile> imply G p\<close>
-proof (rule K_Boole, rule ccontr)
-  assume \<open>\<not> AxB \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close>
+  assumes valid: \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M.
+    symmetric M \<longrightarrow> (\<forall>q \<in> G. M, w \<Turnstile> q) \<longrightarrow> M, w \<Turnstile> p\<close>
+  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (AxB \<turnstile> imply qs p)\<close>
+proof (rule ccontr)
+  assume \<open>\<nexists>qs. set qs \<subseteq> G \<and> AxB \<turnstile> imply qs p\<close>
+  then have *: \<open>\<forall>qs. set qs \<subseteq> G \<longrightarrow> \<not> AxB \<turnstile> imply ((\<^bold>\<not> p) # qs) \<^bold>\<bottom>\<close>
+    using K_Boole by blast
 
-  let ?S = \<open>set ((\<^bold>\<not> p) # G)\<close>
+  let ?S = \<open>{\<^bold>\<not> p} \<union> G\<close>
   let ?V = \<open>Extend AxB ?S from_nat\<close>
-  let ?M = \<open>Kripke pi (reach AxB)\<close>
-  let ?V\<^sub>K\<^sub>B = \<open>Abs_mcs\<^sub>K\<^sub>B ?V\<close>
-  let ?M\<^sub>K\<^sub>B = \<open>Kripke pi\<^sub>K\<^sub>B reach\<^sub>K\<^sub>B\<close>
+  let ?M = \<open>Kripke (mcss AxB) pi (reach AxB)\<close>
 
   have \<open>consistent AxB ?S\<close>
-    unfolding consistent_def using \<open>\<not> AxB \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close> K_imply_weaken by blast
-  then have \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>list_all (\<lambda>p. ?M, ?V \<Turnstile> p) G\<close> \<open>consistent AxB ?V\<close> \<open>maximal AxB ?V\<close>
+    using * by (metis K_imply_Cons consistent_def inconsistent_subset)
+  then have \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>\<forall>q \<in> G. ?M, ?V \<Turnstile> q\<close> \<open>consistent AxB ?V\<close> \<open>maximal AxB ?V\<close>
     using canonical_model unfolding list_all_def by fastforce+
-  then have \<open>?M\<^sub>K\<^sub>B, ?V\<^sub>K\<^sub>B \<Turnstile> (\<^bold>\<not> p)\<close> \<open>list_all (\<lambda>p. ?M\<^sub>K\<^sub>B, ?V\<^sub>K\<^sub>B \<Turnstile> p) G\<close>
-    using \<open>consistent AxB ?V\<close> \<open>maximal AxB ?V\<close> mcs\<^sub>K\<^sub>B_equiv unfolding list_all_def by blast+
-  moreover have \<open>symmetric ?M\<^sub>K\<^sub>B\<close>
+  moreover have \<open>symmetric ?M\<close>
     using mcs\<^sub>K\<^sub>B_symmetric by fast
-  ultimately have \<open>?M\<^sub>K\<^sub>B, ?V\<^sub>K\<^sub>B \<Turnstile> p\<close>
-    using valid by blast
+  ultimately have \<open>?M, ?V \<Turnstile> p\<close>
+    using valid by auto
   then show False
-    using \<open>?M\<^sub>K\<^sub>B, ?V\<^sub>K\<^sub>B \<Turnstile> (\<^bold>\<not> p)\<close> by simp
+    using \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> by simp
 qed
 
 lemma completeness\<^sub>K\<^sub>B:
-  assumes \<open>\<forall>(M :: ('i, ('i :: countable) mcs\<^sub>K\<^sub>B) kripke) w. symmetric M \<longrightarrow> M, w \<Turnstile> p\<close>
+  assumes \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M. symmetric M \<longrightarrow> M, w \<Turnstile> p\<close>
   shows \<open>\<turnstile>\<^sub>K\<^sub>B p\<close>
-  using assms imply_completeness_KB[where G=\<open>[]\<close>] by auto
+  using assms imply_completeness_KB[where G=\<open>{}\<close>] by auto
 
-abbreviation \<open>valid\<^sub>K\<^sub>B p \<equiv> \<forall>(M :: (nat, nat mcs\<^sub>K\<^sub>B) kripke) w. symmetric M \<longrightarrow> M, w \<Turnstile> p\<close>
+abbreviation \<open>valid\<^sub>K\<^sub>B p \<equiv> \<forall>(M :: (nat, nat fm set) kripke). \<forall>w \<in> \<W> M. symmetric M \<longrightarrow> M, w \<Turnstile> p\<close>
 
 theorem main\<^sub>K\<^sub>B: \<open>valid\<^sub>K\<^sub>B p \<longleftrightarrow> \<turnstile>\<^sub>K\<^sub>B p\<close>
   using soundness\<^sub>K\<^sub>B completeness\<^sub>K\<^sub>B by fast
 
 corollary
-  assumes \<open>symmetric M\<close>
+  assumes \<open>symmetric M\<close> \<open>w \<in> \<W> M\<close>
   shows \<open>valid\<^sub>K\<^sub>B p \<longrightarrow> M, w \<Turnstile> p\<close>
   using assms soundness\<^sub>K\<^sub>B completeness\<^sub>K\<^sub>B by fast
 
 section \<open>System K4\<close>
 
 inductive Ax4 :: \<open>'i fm \<Rightarrow> bool\<close> where
   \<open>Ax4 (K i p \<^bold>\<longrightarrow> K i (K i p))\<close>
 
 abbreviation SystemK4 :: \<open>'i fm \<Rightarrow> bool\<close> ("\<turnstile>\<^sub>K\<^sub>4 _" [50] 50) where
   \<open>\<turnstile>\<^sub>K\<^sub>4 p \<equiv> Ax4 \<turnstile> p\<close>
 
-lemma soundness_Ax4: \<open>Ax4 p \<Longrightarrow> transitive M \<Longrightarrow> M, w \<Turnstile> p\<close>
+lemma soundness_Ax4: \<open>Ax4 p \<Longrightarrow> transitive M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
   by (induct p rule: Ax4.induct) (meson pos_introspection)
 
-lemma soundness\<^sub>K\<^sub>4: \<open>\<turnstile>\<^sub>K\<^sub>4 p \<Longrightarrow> transitive M \<Longrightarrow> M, w \<Turnstile> p\<close>
+lemma soundness\<^sub>K\<^sub>4: \<open>\<turnstile>\<^sub>K\<^sub>4 p \<Longrightarrow> transitive M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
   using soundness soundness_Ax4 .
 
-typedef 'i mcs\<^sub>K\<^sub>4 = \<open>{V :: ('i :: countable) fm set. consistent Ax4 V \<and> maximal Ax4 V}\<close>
-  using ex_mcs[where A=Ax4 and P=transitive] soundness_Ax4 ex_equiv_model by fast
-
-abbreviation \<open>pi\<^sub>K\<^sub>4 \<equiv> pi o Rep_mcs\<^sub>K\<^sub>4\<close>
-abbreviation \<open>reach\<^sub>K\<^sub>4 i V \<equiv> Abs_mcs\<^sub>K\<^sub>4 ` (reach Ax4 i (Rep_mcs\<^sub>K\<^sub>4 V))\<close>
-
-lemma mcs\<^sub>K\<^sub>4_equiv:
-  assumes \<open>consistent Ax4 V\<close> \<open>maximal Ax4 V\<close>
-  shows \<open>(Kripke pi (reach Ax4), V \<Turnstile> p) = (Kripke pi\<^sub>K\<^sub>4 reach\<^sub>K\<^sub>4, Abs_mcs\<^sub>K\<^sub>4 V \<Turnstile> p)\<close>
-  using assms by (induct p arbitrary: V) (simp_all add: Abs_mcs\<^sub>K\<^sub>4_inverse)
-
 lemma Ax4_transitive:
-  assumes \<open>\<And>p. Ax4 p \<Longrightarrow> A p\<close> \<open>consistent A V\<close> \<open>maximal A V\<close>
+  assumes \<open>\<forall>p. Ax4 p \<longrightarrow> A p\<close> \<open>consistent A V\<close> \<open>maximal A V\<close>
     and \<open>W \<in> reach A i V\<close> \<open>U \<in> reach A i W\<close>
   shows \<open>U \<in> reach A i V\<close>
 proof -
   have \<open>(K i p \<^bold>\<longrightarrow> K i (K i p)) \<in> V\<close> for p
     using assms(1-3) ax_in_maximal Ax4.intros by metis
   then have \<open>K i (K i p) \<in> V\<close> if \<open>K i p \<in> V\<close> for p
     using that assms(2-3) consequent_in_maximal by blast
   then show ?thesis
     using assms(4-5) by blast
 qed
 
-lemma mcs\<^sub>K\<^sub>4_transitive: \<open>transitive (Kripke pi\<^sub>K\<^sub>4 reach\<^sub>K\<^sub>4)\<close>
+lemma mcs\<^sub>K\<^sub>4_transitive:
+  assumes \<open>\<forall>p. Ax4 p \<longrightarrow> A p\<close>
+  shows \<open>transitive (Kripke (mcss A) pi (reach A))\<close>
   unfolding transitive_def
 proof safe
-  fix i :: \<open>'i :: countable\<close> and V W U :: \<open>'i mcs\<^sub>K\<^sub>4\<close>
-  let ?V = \<open>Rep_mcs\<^sub>K\<^sub>4 V\<close> and ?W = \<open>Rep_mcs\<^sub>K\<^sub>4 W\<close> and ?U = \<open>Rep_mcs\<^sub>K\<^sub>4 U\<close>
-
-  assume \<open>W \<in> \<K> (Kripke pi\<^sub>K\<^sub>4 reach\<^sub>K\<^sub>4) i V\<close> \<open>U \<in> \<K> (Kripke pi\<^sub>K\<^sub>4 reach\<^sub>K\<^sub>4) i W\<close>
-  then have \<open>?W \<in> reach Ax4 i ?V\<close> \<open>?U \<in> reach Ax4 i ?W\<close>
-    using Abs_mcs\<^sub>K\<^sub>4_inverse by auto
-  moreover have \<open>consistent Ax4 ?V\<close> \<open>maximal Ax4 ?V\<close>
-    using Rep_mcs\<^sub>K\<^sub>4 by blast+
-  ultimately have \<open>?U \<in> reach Ax4 i ?V\<close>
-    using Ax4_transitive[where V=\<open>?V\<close>] by blast
-  then have \<open>U \<in> reach\<^sub>K\<^sub>4 i V\<close>
-    using Rep_mcs\<^sub>K\<^sub>4_inverse by (metis (no_types, lifting) image_iff)
-  then show \<open>U \<in> \<K> (Kripke pi\<^sub>K\<^sub>4 reach\<^sub>K\<^sub>4) i V\<close>
+  fix i U V W
+  assume \<open>V \<in> \<W> (Kripke (mcss A) pi (reach A))\<close>
+  then have \<open>consistent A V\<close> \<open>maximal A V\<close>
+    by simp_all
+  moreover assume
+    \<open>W \<in> \<K> (Kripke (mcss A) pi (reach A)) i V\<close>
+    \<open>U \<in> \<K> (Kripke (mcss A) pi (reach A)) i W\<close>
+  ultimately have \<open>U \<in> reach A i V\<close>
+    using Ax4_transitive[where V=V and W=W and U=U] assms by simp
+  then show \<open>U \<in> \<K> (Kripke (mcss A) pi (reach A)) i V\<close>
     by simp
 qed
 
 lemma imply_completeness_K4:
-  assumes valid: \<open>\<forall>(M :: ('i :: countable, 'i mcs\<^sub>K\<^sub>4) kripke) w.
-    transitive M \<longrightarrow> list_all (\<lambda>q. M, w \<Turnstile> q) G \<longrightarrow> M, w \<Turnstile> p\<close>
-  shows \<open>Ax4 \<turnstile> imply G p\<close>
-proof (rule K_Boole, rule ccontr)
-  assume \<open>\<not> Ax4 \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close>
+  assumes valid: \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M.
+    transitive M \<longrightarrow> (\<forall>q \<in> G. M, w \<Turnstile> q) \<longrightarrow> M, w \<Turnstile> p\<close>
+  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (Ax4 \<turnstile> imply qs p)\<close>
+proof (rule ccontr)
+  assume \<open>\<nexists>qs. set qs \<subseteq> G \<and> Ax4 \<turnstile> imply qs p\<close>
+  then have *: \<open>\<forall>qs. set qs \<subseteq> G \<longrightarrow> \<not> Ax4 \<turnstile> imply ((\<^bold>\<not> p) # qs) \<^bold>\<bottom>\<close>
+    using K_Boole by blast
 
-  let ?S = \<open>set ((\<^bold>\<not> p) # G)\<close>
+  let ?S = \<open>{\<^bold>\<not> p} \<union> G\<close>
   let ?V = \<open>Extend Ax4 ?S from_nat\<close>
-  let ?M = \<open>Kripke pi (reach Ax4)\<close>
-  let ?V\<^sub>K\<^sub>4 = \<open>Abs_mcs\<^sub>K\<^sub>4 ?V\<close>
-  let ?M\<^sub>K\<^sub>4 = \<open>Kripke pi\<^sub>K\<^sub>4 reach\<^sub>K\<^sub>4\<close>
+  let ?M = \<open>Kripke (mcss Ax4) pi (reach Ax4)\<close>
 
   have \<open>consistent Ax4 ?S\<close>
-    unfolding consistent_def using \<open>\<not> Ax4 \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close> K_imply_weaken by blast
-  then have \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>list_all (\<lambda>p. ?M, ?V \<Turnstile> p) G\<close> \<open>consistent Ax4 ?V\<close> \<open>maximal Ax4 ?V\<close>
+    using * by (metis K_imply_Cons consistent_def inconsistent_subset)
+  then have \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>\<forall>q \<in> G. ?M, ?V \<Turnstile> q\<close> \<open>consistent Ax4 ?V\<close> \<open>maximal Ax4 ?V\<close>
     using canonical_model unfolding list_all_def by fastforce+
-  then have \<open>?M\<^sub>K\<^sub>4, ?V\<^sub>K\<^sub>4 \<Turnstile> (\<^bold>\<not> p)\<close> \<open>list_all (\<lambda>p. ?M\<^sub>K\<^sub>4, ?V\<^sub>K\<^sub>4 \<Turnstile> p) G\<close>
-    using \<open>consistent Ax4 ?V\<close> \<open>maximal Ax4 ?V\<close> mcs\<^sub>K\<^sub>4_equiv unfolding list_all_def by blast+
-  moreover have \<open>transitive ?M\<^sub>K\<^sub>4\<close>
+  moreover have \<open>transitive ?M\<close>
     using mcs\<^sub>K\<^sub>4_transitive by fast
-  ultimately have \<open>?M\<^sub>K\<^sub>4, ?V\<^sub>K\<^sub>4 \<Turnstile> p\<close>
-    using valid by blast
+  ultimately have \<open>?M, ?V \<Turnstile> p\<close>
+    using valid by auto
   then show False
-    using \<open>?M\<^sub>K\<^sub>4, ?V\<^sub>K\<^sub>4 \<Turnstile> (\<^bold>\<not> p)\<close> by simp
+    using \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> by simp
 qed
 
 lemma completeness\<^sub>K\<^sub>4:
-  assumes \<open>\<forall>(M :: ('i, ('i :: countable) mcs\<^sub>K\<^sub>4) kripke) w. transitive M \<longrightarrow> M, w \<Turnstile> p\<close>
+  assumes \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M. transitive M \<longrightarrow> M, w \<Turnstile> p\<close>
   shows \<open>\<turnstile>\<^sub>K\<^sub>4 p\<close>
-  using assms imply_completeness_K4[where G=\<open>[]\<close>] by auto
+  using assms imply_completeness_K4[where G=\<open>{}\<close>] by auto
 
-abbreviation \<open>valid\<^sub>K\<^sub>4 p \<equiv> \<forall>(M :: (nat, nat mcs\<^sub>K\<^sub>4) kripke) w. transitive M \<longrightarrow> M, w \<Turnstile> p\<close>
+abbreviation \<open>valid\<^sub>K\<^sub>4 p \<equiv> \<forall>(M :: (nat, nat fm set) kripke). \<forall>w \<in> \<W> M. transitive M \<longrightarrow> M, w \<Turnstile> p\<close>
 
 theorem main\<^sub>K\<^sub>4: \<open>valid\<^sub>K\<^sub>4 p \<longleftrightarrow> \<turnstile>\<^sub>K\<^sub>4 p\<close>
   using soundness\<^sub>K\<^sub>4 completeness\<^sub>K\<^sub>4 by fast
 
 corollary
-  assumes \<open>transitive M\<close>
+  assumes \<open>transitive M\<close> \<open>w \<in> \<W> M\<close>
   shows \<open>valid\<^sub>K\<^sub>4 p \<longrightarrow> M, w \<Turnstile> p\<close>
   using assms soundness\<^sub>K\<^sub>4 completeness\<^sub>K\<^sub>4 by fast
 
 section \<open>System S4\<close>
 
 abbreviation Or :: \<open>('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool\<close> (infixl \<open>\<oplus>\<close> 65) where
   \<open>A \<oplus> A' \<equiv> \<lambda>x. A x \<or> A' x\<close>
 
 abbreviation SystemS4 :: \<open>'i fm \<Rightarrow> bool\<close> ("\<turnstile>\<^sub>S\<^sub>4 _" [50] 50) where
   \<open>\<turnstile>\<^sub>S\<^sub>4 p \<equiv> AxT \<oplus> Ax4 \<turnstile> p\<close>
 
-lemma soundness_AxT4: \<open>(AxT \<oplus> Ax4) p \<Longrightarrow> reflexive M \<and> transitive M \<Longrightarrow> M, w \<Turnstile> p\<close>
+lemma soundness_AxT4: \<open>(AxT \<oplus> Ax4) p \<Longrightarrow> reflexive M \<and> transitive M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
   using soundness_AxT soundness_Ax4 by fast
 
-lemma soundness\<^sub>S\<^sub>4: \<open>\<turnstile>\<^sub>S\<^sub>4 p \<Longrightarrow> reflexive M \<and> transitive M \<Longrightarrow> M, w \<Turnstile> p\<close>
+lemma soundness\<^sub>S\<^sub>4: \<open>\<turnstile>\<^sub>S\<^sub>4 p \<Longrightarrow> reflexive M \<and> transitive M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
   using soundness soundness_AxT4 .
 
-typedef 'i mcs\<^sub>S\<^sub>4 =
-  \<open>{V :: ('i :: countable) fm set. consistent (AxT \<oplus> Ax4) V \<and> maximal (AxT \<oplus> Ax4) V}\<close>
-  using ex_mcs[where A=\<open>AxT \<oplus> Ax4\<close> and P=\<open>\<lambda>M. reflexive M \<and> transitive M\<close>]
-    soundness_AxT4 ex_equiv_model by fast
-
-abbreviation \<open>pi\<^sub>S\<^sub>4 \<equiv> pi o Rep_mcs\<^sub>S\<^sub>4\<close>
-abbreviation \<open>reach\<^sub>S\<^sub>4 i V \<equiv> Abs_mcs\<^sub>S\<^sub>4 ` (reach (AxT \<oplus> Ax4) i (Rep_mcs\<^sub>S\<^sub>4 V))\<close>
-
-lemma mcs\<^sub>S\<^sub>4_equiv:
-  assumes \<open>consistent (AxT \<oplus> Ax4) V\<close> \<open>maximal (AxT \<oplus> Ax4) V\<close>
-  shows \<open>(Kripke pi (reach (AxT \<oplus> Ax4)), V \<Turnstile> p) = (Kripke pi\<^sub>S\<^sub>4 reach\<^sub>S\<^sub>4, Abs_mcs\<^sub>S\<^sub>4 V \<Turnstile> p)\<close>
-  using assms by (induct p arbitrary: V) (simp_all add: Abs_mcs\<^sub>S\<^sub>4_inverse)
-
-lemma mcs\<^sub>S\<^sub>4_reflexive: \<open>reflexive (Kripke pi\<^sub>S\<^sub>4 reach\<^sub>S\<^sub>4)\<close>
-  unfolding reflexive_def
-proof (intro allI)
-  fix i :: \<open>'i :: countable\<close> and V :: \<open>'i mcs\<^sub>S\<^sub>4\<close>
-  let ?V = \<open>Rep_mcs\<^sub>S\<^sub>4 V\<close>
-  have \<open>consistent (AxT \<oplus> Ax4) ?V\<close> \<open>maximal (AxT \<oplus> Ax4) ?V\<close>
-    using Rep_mcs\<^sub>S\<^sub>4 by blast+
-  then have \<open>?V \<in> reach (AxT \<oplus> Ax4) i ?V\<close>
-    using AxT_reflexive[where V=\<open>?V\<close>] by presburger
-  then have \<open>V \<in> reach\<^sub>S\<^sub>4 i V\<close>
-    using Rep_mcs\<^sub>S\<^sub>4_inverse by (metis (no_types, lifting) image_iff)
-  then show \<open>V \<in> \<K> (Kripke pi\<^sub>S\<^sub>4 reach\<^sub>S\<^sub>4) i V\<close>
-    by simp
-qed
+lemma imply_completeness_S4:
+  assumes valid: \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M.
+    reflexive M \<longrightarrow> transitive M \<longrightarrow> (\<forall>q \<in> G. M, w \<Turnstile> q) \<longrightarrow> M, w \<Turnstile> p\<close>
+  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (AxT \<oplus> Ax4 \<turnstile> imply qs p)\<close>
+proof (rule ccontr)
+  assume \<open>\<nexists>qs. set qs \<subseteq> G \<and> AxT \<oplus> Ax4 \<turnstile> imply qs p\<close>
+  then have *: \<open>\<forall>qs. set qs \<subseteq> G \<longrightarrow> \<not> AxT \<oplus> Ax4 \<turnstile> imply ((\<^bold>\<not> p) # qs) \<^bold>\<bottom>\<close>
+    using K_Boole by blast
 
-lemma mcs\<^sub>S\<^sub>4_transitive: \<open>transitive (Kripke pi\<^sub>S\<^sub>4 reach\<^sub>S\<^sub>4)\<close>
-  unfolding transitive_def
-proof safe
-  fix i :: \<open>'i :: countable\<close> and V W U :: \<open>'i mcs\<^sub>S\<^sub>4\<close>
-  let ?V = \<open>Rep_mcs\<^sub>S\<^sub>4 V\<close> and ?W = \<open>Rep_mcs\<^sub>S\<^sub>4 W\<close> and ?U = \<open>Rep_mcs\<^sub>S\<^sub>4 U\<close>
-
-  assume \<open>W \<in> \<K> (Kripke pi\<^sub>S\<^sub>4 reach\<^sub>S\<^sub>4) i V\<close> \<open>U \<in> \<K> (Kripke pi\<^sub>S\<^sub>4 reach\<^sub>S\<^sub>4) i W\<close>
-  then have \<open>?W \<in> reach (AxT \<oplus> Ax4) i ?V\<close> \<open>?U \<in> reach (AxT \<oplus> Ax4) i ?W\<close>
-    using Abs_mcs\<^sub>S\<^sub>4_inverse by auto
-  moreover have \<open>consistent (AxT \<oplus> Ax4) ?V\<close> \<open>maximal (AxT \<oplus> Ax4) ?V\<close>
-    using Rep_mcs\<^sub>S\<^sub>4 by blast+
-  ultimately have \<open>?U \<in> reach (AxT \<oplus> Ax4) i ?V\<close>
-    using Ax4_transitive[where V=\<open>?V\<close>] by presburger
-  then have \<open>U \<in> reach\<^sub>S\<^sub>4 i V\<close>
-    using Rep_mcs\<^sub>S\<^sub>4_inverse by (metis (no_types, lifting) image_iff)
-  then show \<open>U \<in> \<K> (Kripke pi\<^sub>S\<^sub>4 reach\<^sub>S\<^sub>4) i V\<close>
-    by simp
-qed
-
-lemma imply_completeness_S4:
-  assumes valid: \<open>\<forall>(M :: ('i :: countable, 'i mcs\<^sub>S\<^sub>4) kripke) w.
-    reflexive M \<longrightarrow> transitive M \<longrightarrow> list_all (\<lambda>q. M, w \<Turnstile> q) G \<longrightarrow> M, w \<Turnstile> p\<close>
-  shows \<open>AxT \<oplus> Ax4 \<turnstile> imply G p\<close>
-proof (rule K_Boole, rule ccontr)
-  assume \<open>\<not> AxT \<oplus> Ax4 \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close>
-
-  let ?S = \<open>set ((\<^bold>\<not> p) # G)\<close>
+  let ?S = \<open>{\<^bold>\<not> p} \<union> G\<close>
   let ?V = \<open>Extend (AxT \<oplus> Ax4) ?S from_nat\<close>
-  let ?M = \<open>Kripke pi (reach (AxT \<oplus> Ax4))\<close>
-  let ?V\<^sub>S\<^sub>4 = \<open>Abs_mcs\<^sub>S\<^sub>4 ?V\<close>
-  let ?M\<^sub>S\<^sub>4 = \<open>Kripke pi\<^sub>S\<^sub>4 reach\<^sub>S\<^sub>4\<close>
+  let ?M = \<open>Kripke (mcss (AxT \<oplus> Ax4)) pi (reach (AxT \<oplus> Ax4))\<close>
 
   have \<open>consistent (AxT \<oplus> Ax4) ?S\<close>
-    unfolding consistent_def using \<open>\<not> AxT \<oplus> Ax4 \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close> K_imply_weaken by blast
+    using * by (metis (no_types, lifting) K_imply_Cons consistent_def inconsistent_subset)
   then have
-    \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>list_all (\<lambda>p. ?M, ?V \<Turnstile> p) G\<close>
+    \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>\<forall>q \<in> G. ?M, ?V \<Turnstile> q\<close>
     \<open>consistent (AxT \<oplus> Ax4) ?V\<close> \<open>maximal (AxT \<oplus> Ax4) ?V\<close>
     using canonical_model unfolding list_all_def by fastforce+
-  then have \<open>?M\<^sub>S\<^sub>4, ?V\<^sub>S\<^sub>4 \<Turnstile> (\<^bold>\<not> p)\<close> \<open>list_all (\<lambda>p. ?M\<^sub>S\<^sub>4, ?V\<^sub>S\<^sub>4 \<Turnstile> p) G\<close>
-    using \<open>consistent (AxT \<oplus> Ax4) ?V\<close> \<open>maximal (AxT \<oplus> Ax4) ?V\<close> mcs\<^sub>S\<^sub>4_equiv
-    unfolding list_all_def by blast+
-  moreover have \<open>reflexive ?M\<^sub>S\<^sub>4\<close> \<open>transitive ?M\<^sub>S\<^sub>4\<close>
-    using mcs\<^sub>S\<^sub>4_reflexive mcs\<^sub>S\<^sub>4_transitive by fast+
-  ultimately have \<open>?M\<^sub>S\<^sub>4, ?V\<^sub>S\<^sub>4 \<Turnstile> p\<close>
-    using valid by blast
+  moreover have \<open>reflexive ?M\<close> \<open>transitive ?M\<close>
+    by (simp_all add: mcs\<^sub>T_reflexive mcs\<^sub>K\<^sub>4_transitive)
+  ultimately have \<open>?M, ?V \<Turnstile> p\<close>
+    using valid by auto
   then show False
-    using \<open>?M\<^sub>S\<^sub>4, ?V\<^sub>S\<^sub>4 \<Turnstile> (\<^bold>\<not> p)\<close> by simp
+    using \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> by simp
 qed
 
 lemma completeness\<^sub>S\<^sub>4:
-  assumes \<open>\<forall>(M :: ('i, ('i :: countable) mcs\<^sub>S\<^sub>4) kripke) w.
+  assumes \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M.
       reflexive M \<longrightarrow> transitive M \<longrightarrow> M, w \<Turnstile> p\<close>
   shows \<open>\<turnstile>\<^sub>S\<^sub>4 p\<close>
-  using assms imply_completeness_S4[where G=\<open>[]\<close>] by auto
+  using assms imply_completeness_S4[where G=\<open>{}\<close>] by auto
 
-abbreviation \<open>valid\<^sub>S\<^sub>4 p \<equiv> \<forall>(M :: (nat, nat mcs\<^sub>S\<^sub>4) kripke) w.
+abbreviation \<open>valid\<^sub>S\<^sub>4 p \<equiv> \<forall>(M :: (nat, nat fm set) kripke). \<forall>w \<in> \<W> M.
     reflexive M \<longrightarrow> transitive M \<longrightarrow> M, w \<Turnstile> p\<close>
 
 theorem main\<^sub>S\<^sub>4: \<open>valid\<^sub>S\<^sub>4 p \<longleftrightarrow> \<turnstile>\<^sub>S\<^sub>4 p\<close>
   using soundness\<^sub>S\<^sub>4 completeness\<^sub>S\<^sub>4 by fast
 
 corollary
-  assumes \<open>reflexive M\<close> \<open>transitive M\<close>
+  assumes \<open>reflexive M\<close> \<open>transitive M\<close> \<open>w \<in> \<W> M\<close>
   shows \<open>valid\<^sub>S\<^sub>4 p \<longrightarrow> M, w \<Turnstile> p\<close>
   using assms soundness\<^sub>S\<^sub>4 completeness\<^sub>S\<^sub>4 by fast
 
 section \<open>System S5\<close>
 
 abbreviation SystemS5 :: \<open>'i fm \<Rightarrow> bool\<close> ("\<turnstile>\<^sub>S\<^sub>5 _" [50] 50) where
   \<open>\<turnstile>\<^sub>S\<^sub>5 p \<equiv> AxT \<oplus> AxB \<oplus> Ax4 \<turnstile> p\<close>
 
 abbreviation AxTB4 :: \<open>'i fm \<Rightarrow> bool\<close> where
   \<open>AxTB4 \<equiv> AxT \<oplus> AxB \<oplus> Ax4\<close>
 
-lemma soundness_AxTB4: \<open>AxTB4 p \<Longrightarrow> equivalence M \<Longrightarrow> M, w \<Turnstile> p\<close>
+lemma soundness_AxTB4: \<open>AxTB4 p \<Longrightarrow> equivalence M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
   using soundness_AxT soundness_AxB soundness_Ax4 by fast
 
-lemma soundness\<^sub>S\<^sub>5: \<open>\<turnstile>\<^sub>S\<^sub>5 p \<Longrightarrow> equivalence M \<Longrightarrow> M, w \<Turnstile> p\<close>
+lemma soundness\<^sub>S\<^sub>5: \<open>\<turnstile>\<^sub>S\<^sub>5 p \<Longrightarrow> equivalence M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
   using soundness soundness_AxTB4 .
 
-typedef 'i mcs\<^sub>S\<^sub>5 =
-  \<open>{V :: ('i :: countable) fm set. consistent AxTB4 V \<and> maximal AxTB4 V}\<close>
-  using ex_mcs[where A=AxTB4 and P=equivalence] soundness_AxTB4 ex_equiv_model by fast
-
-abbreviation \<open>pi\<^sub>S\<^sub>5 \<equiv> pi o Rep_mcs\<^sub>S\<^sub>5\<close>
-abbreviation \<open>reach\<^sub>S\<^sub>5 i V \<equiv> Abs_mcs\<^sub>S\<^sub>5 ` (reach AxTB4 i (Rep_mcs\<^sub>S\<^sub>5 V))\<close>
-
-lemma mcs\<^sub>S\<^sub>5_equiv:
-  assumes \<open>consistent AxTB4 V\<close> \<open>maximal AxTB4 V\<close>
-  shows \<open>(Kripke pi (reach AxTB4), V \<Turnstile> p) = (Kripke pi\<^sub>S\<^sub>5 reach\<^sub>S\<^sub>5, Abs_mcs\<^sub>S\<^sub>5 V \<Turnstile> p)\<close>
-  using assms by (induct p arbitrary: V) (simp_all add: Abs_mcs\<^sub>S\<^sub>5_inverse)
-
-lemma mcs\<^sub>S\<^sub>5_reflexive: \<open>reflexive (Kripke pi\<^sub>S\<^sub>5 reach\<^sub>S\<^sub>5)\<close>
-  unfolding reflexive_def
-proof (intro allI)
-  fix i :: \<open>'i :: countable\<close> and V :: \<open>'i mcs\<^sub>S\<^sub>5\<close>
-  let ?V = \<open>Rep_mcs\<^sub>S\<^sub>5 V\<close>
-  have \<open>consistent AxTB4 ?V\<close> \<open>maximal AxTB4 ?V\<close>
-    using Rep_mcs\<^sub>S\<^sub>5 by blast+
-  then have \<open>?V \<in> reach AxTB4 i ?V\<close>
-    using AxT_reflexive[where V=\<open>?V\<close>] by presburger
-  then have \<open>V \<in> reach\<^sub>S\<^sub>5 i V\<close>
-    using Rep_mcs\<^sub>S\<^sub>5_inverse by (metis (no_types, lifting) image_iff)
-  then show \<open>V \<in> \<K> (Kripke pi\<^sub>S\<^sub>5 reach\<^sub>S\<^sub>5) i V\<close>
-    by simp
-qed
-
-lemma mcs\<^sub>S\<^sub>5_symmetric: \<open>symmetric (Kripke pi\<^sub>S\<^sub>5 reach\<^sub>S\<^sub>5)\<close>
-  unfolding symmetric_def
-proof (intro allI)
-  fix i :: \<open>'i :: countable\<close> and V W :: \<open>'i mcs\<^sub>S\<^sub>5\<close>
-  let ?V = \<open>Rep_mcs\<^sub>S\<^sub>5 V\<close> and ?W = \<open>Rep_mcs\<^sub>S\<^sub>5 W\<close>
+lemma imply_completeness_S5:
+  assumes valid: \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M.
+    equivalence M \<longrightarrow> (\<forall>q \<in> G. M, w \<Turnstile> q) \<longrightarrow> M, w \<Turnstile> p\<close>
+  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (AxTB4 \<turnstile> imply qs p)\<close>
+proof (rule ccontr)
+  assume \<open>\<nexists>qs. set qs \<subseteq> G \<and> AxTB4 \<turnstile> imply qs p\<close>
+  then have *: \<open>\<forall>qs. set qs \<subseteq> G \<longrightarrow> \<not> AxTB4 \<turnstile> imply ((\<^bold>\<not> p) # qs) \<^bold>\<bottom>\<close>
+    using K_Boole by blast
 
-  have \<open>consistent AxTB4 ?V\<close> \<open>maximal AxTB4 ?V\<close>
-    \<open>consistent AxTB4 ?W\<close> \<open>maximal AxTB4 ?W\<close>
-    using Rep_mcs\<^sub>S\<^sub>5 by blast+
-  then have \<open>?V \<in> reach AxTB4 i ?W \<longleftrightarrow> ?W \<in> reach AxTB4 i ?V\<close>
-    using AxB_symmetric[where V=\<open>?V\<close> and W=\<open>?W\<close>] by presburger
-  then have \<open>V \<in> reach\<^sub>S\<^sub>5 i W \<longleftrightarrow> W \<in> reach\<^sub>S\<^sub>5 i V\<close>
-    unfolding image_iff using Rep_mcs\<^sub>S\<^sub>5_cases Rep_mcs\<^sub>S\<^sub>5_inverse
-    by (metis (mono_tags, lifting) mem_Collect_eq)
-  then show \<open>(W \<in> \<K> (Kripke pi\<^sub>S\<^sub>5 reach\<^sub>S\<^sub>5) i V) = (V \<in> \<K> (Kripke pi\<^sub>S\<^sub>5 reach\<^sub>S\<^sub>5) i W)\<close>
-    by simp
-qed
-
-lemma mcs\<^sub>S\<^sub>5_transitive: \<open>transitive (Kripke pi\<^sub>S\<^sub>5 reach\<^sub>S\<^sub>5)\<close>
-  unfolding transitive_def
-proof safe
-  fix i :: \<open>'i :: countable\<close> and V W U :: \<open>'i mcs\<^sub>S\<^sub>5\<close>
-  let ?V = \<open>Rep_mcs\<^sub>S\<^sub>5 V\<close> and ?W = \<open>Rep_mcs\<^sub>S\<^sub>5 W\<close> and ?U = \<open>Rep_mcs\<^sub>S\<^sub>5 U\<close>
-
-  assume \<open>W \<in> \<K> (Kripke pi\<^sub>S\<^sub>5 reach\<^sub>S\<^sub>5) i V\<close> \<open>U \<in> \<K> (Kripke pi\<^sub>S\<^sub>5 reach\<^sub>S\<^sub>5) i W\<close>
-  then have \<open>?W \<in> reach AxTB4 i ?V\<close> \<open>?U \<in> reach AxTB4 i ?W\<close>
-    using Abs_mcs\<^sub>S\<^sub>5_inverse by auto
-  moreover have \<open>consistent AxTB4 ?V\<close> \<open>maximal AxTB4 ?V\<close>
-    using Rep_mcs\<^sub>S\<^sub>5 by blast+
-  ultimately have \<open>?U \<in> reach AxTB4 i ?V\<close>
-    using Ax4_transitive[where V=\<open>?V\<close>] by presburger
-  then have \<open>U \<in> reach\<^sub>S\<^sub>5 i V\<close>
-    using Rep_mcs\<^sub>S\<^sub>5_inverse by (metis (no_types, lifting) image_iff)
-  then show \<open>U \<in> \<K> (Kripke pi\<^sub>S\<^sub>5 reach\<^sub>S\<^sub>5) i V\<close>
-    by simp
-qed
-
-lemma imply_completeness_S5:
-  assumes valid: \<open>\<forall>(M :: ('i :: countable, 'i mcs\<^sub>S\<^sub>5) kripke) w.
-    equivalence M \<longrightarrow> list_all (\<lambda>q. M, w \<Turnstile> q) G \<longrightarrow> M, w \<Turnstile> p\<close>
-  shows \<open>AxTB4 \<turnstile> imply G p\<close>
-proof (rule K_Boole, rule ccontr)
-  assume \<open>\<not> AxTB4 \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close>
-
-  let ?S = \<open>set ((\<^bold>\<not> p) # G)\<close>
+  let ?S = \<open>{\<^bold>\<not> p} \<union> G\<close>
   let ?V = \<open>Extend AxTB4 ?S from_nat\<close>
-  let ?M = \<open>Kripke pi (reach AxTB4)\<close>
-  let ?V\<^sub>S\<^sub>5 = \<open>Abs_mcs\<^sub>S\<^sub>5 ?V\<close>
-  let ?M\<^sub>S\<^sub>5 = \<open>Kripke pi\<^sub>S\<^sub>5 reach\<^sub>S\<^sub>5\<close>
+  let ?M = \<open>Kripke (mcss AxTB4) pi (reach AxTB4)\<close>
 
   have \<open>consistent AxTB4 ?S\<close>
-    unfolding consistent_def using \<open>\<not> AxTB4 \<turnstile> imply ((\<^bold>\<not> p) # G) \<^bold>\<bottom>\<close> K_imply_weaken by blast
+    using * by (metis (no_types, lifting) K_imply_Cons consistent_def inconsistent_subset)
   then have
-    \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>list_all (\<lambda>p. ?M, ?V \<Turnstile> p) G\<close>
+    \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> \<open>\<forall>q \<in> G. ?M, ?V \<Turnstile> q\<close>
     \<open>consistent AxTB4 ?V\<close> \<open>maximal AxTB4 ?V\<close>
     using canonical_model unfolding list_all_def by fastforce+
-  then have \<open>?M\<^sub>S\<^sub>5, ?V\<^sub>S\<^sub>5 \<Turnstile> (\<^bold>\<not> p)\<close> \<open>list_all (\<lambda>p. ?M\<^sub>S\<^sub>5, ?V\<^sub>S\<^sub>5 \<Turnstile> p) G\<close>
-    using \<open>consistent AxTB4 ?V\<close> \<open>maximal AxTB4 ?V\<close> mcs\<^sub>S\<^sub>5_equiv
-    unfolding list_all_def by blast+
-  moreover have \<open>equivalence ?M\<^sub>S\<^sub>5\<close>
-    using mcs\<^sub>S\<^sub>5_reflexive mcs\<^sub>S\<^sub>5_symmetric mcs\<^sub>S\<^sub>5_transitive by fast
-  ultimately have \<open>?M\<^sub>S\<^sub>5, ?V\<^sub>S\<^sub>5 \<Turnstile> p\<close>
-    using valid by blast
+  moreover have \<open>equivalence ?M\<close>
+    by (simp add: mcs\<^sub>T_reflexive mcs\<^sub>K\<^sub>B_symmetric mcs\<^sub>K\<^sub>4_transitive)
+  ultimately have \<open>?M, ?V \<Turnstile> p\<close>
+    using valid by auto
   then show False
-    using \<open>?M\<^sub>S\<^sub>5, ?V\<^sub>S\<^sub>5 \<Turnstile> (\<^bold>\<not> p)\<close> by simp
+    using \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> by simp
 qed
 
 lemma completeness\<^sub>S\<^sub>5:
-  assumes \<open>\<forall>(M :: ('i, ('i :: countable) mcs\<^sub>S\<^sub>5) kripke) w. equivalence M \<longrightarrow> M, w \<Turnstile> p\<close>
+  assumes \<open>\<forall>(M :: ('i :: countable, 'i fm set) kripke). \<forall>w \<in> \<W> M. equivalence M \<longrightarrow> M, w \<Turnstile> p\<close>
   shows \<open>\<turnstile>\<^sub>S\<^sub>5 p\<close>
-  using assms imply_completeness_S5[where G=\<open>[]\<close>] by auto
+  using assms imply_completeness_S5[where G=\<open>{}\<close>] by auto
 
-abbreviation \<open>valid\<^sub>S\<^sub>5 p \<equiv> \<forall>(M :: (nat, nat mcs\<^sub>S\<^sub>5) kripke) w. equivalence M \<longrightarrow> M, w \<Turnstile> p\<close>
+abbreviation \<open>valid\<^sub>S\<^sub>5 p \<equiv> \<forall>(M :: (nat, nat fm set) kripke). \<forall>w \<in> \<W> M. equivalence M \<longrightarrow> M, w \<Turnstile> p\<close>
 
 theorem main\<^sub>S\<^sub>5: \<open>valid\<^sub>S\<^sub>5 p \<longleftrightarrow> \<turnstile>\<^sub>S\<^sub>5 p\<close>
   using soundness\<^sub>S\<^sub>5 completeness\<^sub>S\<^sub>5 by fast
 
 corollary
-  assumes \<open>equivalence M\<close>
+  assumes \<open>equivalence M\<close> \<open>w \<in> \<W> M\<close>
   shows \<open>valid\<^sub>S\<^sub>5 p \<longrightarrow> M, w \<Turnstile> p\<close>
   using assms soundness\<^sub>S\<^sub>5 completeness\<^sub>S\<^sub>5 by fast
 
 subsection \<open>Traditional formulation\<close>
 
 inductive SystemS5' :: \<open>'i fm \<Rightarrow> bool\<close> ("\<turnstile>\<^sub>S\<^sub>5'' _" [50] 50) where
   A1': \<open>tautology p \<Longrightarrow> \<turnstile>\<^sub>S\<^sub>5' p\<close>
 | A2': \<open>\<turnstile>\<^sub>S\<^sub>5' (K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i p \<^bold>\<longrightarrow> K i q)\<close>
 | AT': \<open>\<turnstile>\<^sub>S\<^sub>5' (K i p \<^bold>\<longrightarrow> p)\<close>
 | A5': \<open>\<turnstile>\<^sub>S\<^sub>5' (\<^bold>\<not> K i p \<^bold>\<longrightarrow> K i (\<^bold>\<not> K i p))\<close>
 | R1': \<open>\<turnstile>\<^sub>S\<^sub>5' p \<Longrightarrow> \<turnstile>\<^sub>S\<^sub>5' (p \<^bold>\<longrightarrow> q) \<Longrightarrow> \<turnstile>\<^sub>S\<^sub>5' q\<close>
 | R2': \<open>\<turnstile>\<^sub>S\<^sub>5' p \<Longrightarrow> \<turnstile>\<^sub>S\<^sub>5' K i p\<close>
 
 lemma S5'_trans: \<open>\<turnstile>\<^sub>S\<^sub>5' ((p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> (q \<^bold>\<longrightarrow> r) \<^bold>\<longrightarrow> p \<^bold>\<longrightarrow> r)\<close>
   by (simp add: A1')
 
 lemma S5'_L: \<open>\<turnstile>\<^sub>S\<^sub>5' (p \<^bold>\<longrightarrow> L i p)\<close>
 proof -
   have \<open>\<turnstile>\<^sub>S\<^sub>5' (K i (\<^bold>\<not> p) \<^bold>\<longrightarrow> \<^bold>\<not> p)\<close>
     using AT' by fast
   moreover have \<open>\<turnstile>\<^sub>S\<^sub>5' ((P \<^bold>\<longrightarrow> \<^bold>\<not> Q) \<^bold>\<longrightarrow> Q \<^bold>\<longrightarrow> \<^bold>\<not> P)\<close> for P Q :: \<open>'i fm\<close>
     using A1' by force
   ultimately show ?thesis
     using R1' by blast
 qed
 
 lemma S5'_B: \<open>\<turnstile>\<^sub>S\<^sub>5' (p \<^bold>\<longrightarrow> K i (L i p))\<close>
   using A5' S5'_L R1' S5'_trans by metis
 
 lemma S5'_KL: \<open>\<turnstile>\<^sub>S\<^sub>5' (K i p \<^bold>\<longrightarrow> L i p)\<close>
   by (meson AT' R1' S5'_L S5'_trans)
 
 lemma S5'_map_K:
   assumes \<open>\<turnstile>\<^sub>S\<^sub>5' (p \<^bold>\<longrightarrow> q)\<close>
   shows \<open>\<turnstile>\<^sub>S\<^sub>5' (K i p \<^bold>\<longrightarrow> K i q)\<close>
 proof -
   note \<open>\<turnstile>\<^sub>S\<^sub>5' (p \<^bold>\<longrightarrow> q)\<close>
   then have \<open>\<turnstile>\<^sub>S\<^sub>5' K i (p \<^bold>\<longrightarrow> q)\<close>
     using R2' by fast
   moreover have \<open>\<turnstile>\<^sub>S\<^sub>5' (K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i p \<^bold>\<longrightarrow> K i q)\<close>
     using A2' by fast
   ultimately show ?thesis
     using R1' by fast
 qed
 
 lemma S5'_map_L:
   assumes \<open>\<turnstile>\<^sub>S\<^sub>5' (p \<^bold>\<longrightarrow> q)\<close>
   shows \<open>\<turnstile>\<^sub>S\<^sub>5' (L i p \<^bold>\<longrightarrow> L i q)\<close>
   using assms by (metis R1' S5'_map_K S5'_trans)
 
 lemma S5'_L_dual: \<open>\<turnstile>\<^sub>S\<^sub>5' (\<^bold>\<not> L i (\<^bold>\<not> p) \<^bold>\<longrightarrow> K i p)\<close>
 proof -
   have \<open>\<turnstile>\<^sub>S\<^sub>5' (K i p \<^bold>\<longrightarrow> K i p)\<close> \<open>\<turnstile>\<^sub>S\<^sub>5' (\<^bold>\<not> \<^bold>\<not> p \<^bold>\<longrightarrow> p)\<close>
     by (simp_all add: A1')
   then have \<open>\<turnstile>\<^sub>S\<^sub>5' (K i (\<^bold>\<not> \<^bold>\<not> p) \<^bold>\<longrightarrow> K i p)\<close>
     by (simp add: S5'_map_K)
   moreover have \<open>\<turnstile>\<^sub>S\<^sub>5' ((P \<^bold>\<longrightarrow> Q) \<^bold>\<longrightarrow> (\<^bold>\<not> \<^bold>\<not> P \<^bold>\<longrightarrow> Q))\<close> for P Q :: \<open>'i fm\<close>
     by (simp add: A1')
   ultimately show \<open>\<turnstile>\<^sub>S\<^sub>5' (\<^bold>\<not> \<^bold>\<not> K i (\<^bold>\<not> \<^bold>\<not> p) \<^bold>\<longrightarrow> K i p)\<close>
     using R1' by blast
 qed
 
 lemma S5'_4: \<open>\<turnstile>\<^sub>S\<^sub>5' (K i p \<^bold>\<longrightarrow> K i (K i p))\<close>
 proof -
   have \<open>\<turnstile>\<^sub>S\<^sub>5' (L i (\<^bold>\<not> p) \<^bold>\<longrightarrow> K i (L i (\<^bold>\<not> p)))\<close>
     using A5' by fast
   moreover have \<open>\<turnstile>\<^sub>S\<^sub>5' ((P \<^bold>\<longrightarrow> Q) \<^bold>\<longrightarrow> \<^bold>\<not> Q \<^bold>\<longrightarrow> \<^bold>\<not> P)\<close> for P Q :: \<open>'i fm\<close>
     using A1' by force
   ultimately have \<open>\<turnstile>\<^sub>S\<^sub>5' (\<^bold>\<not> K i (L i (\<^bold>\<not> p)) \<^bold>\<longrightarrow> \<^bold>\<not> L i (\<^bold>\<not> p))\<close>
     using R1' by fast
   then have \<open>\<turnstile>\<^sub>S\<^sub>5' (L i (K i (\<^bold>\<not> \<^bold>\<not> p)) \<^bold>\<longrightarrow> \<^bold>\<not> L i (\<^bold>\<not> p))\<close>
     by blast
   moreover have \<open>\<turnstile>\<^sub>S\<^sub>5' (p \<^bold>\<longrightarrow> \<^bold>\<not> \<^bold>\<not> p)\<close>
     by (simp add: A1')
   ultimately have \<open>\<turnstile>\<^sub>S\<^sub>5' (L i (K i p) \<^bold>\<longrightarrow> \<^bold>\<not> L i (\<^bold>\<not> p))\<close>
     by (metis (no_types, opaque_lifting) R1' S5'_map_K S5'_trans)
   then have \<open>\<turnstile>\<^sub>S\<^sub>5' (L i (K i p) \<^bold>\<longrightarrow> K i p)\<close>
     by (meson S5'_L_dual R1' S5'_trans)
   then show ?thesis
     by (metis A2' R1' R2' S5'_B S5'_trans)
 qed
 
 lemma S5_S5': \<open>\<turnstile>\<^sub>S\<^sub>5 p \<Longrightarrow> \<turnstile>\<^sub>S\<^sub>5' p\<close>
 proof (induct p rule: AK.induct)
   case (A2 i p q)
   have \<open>\<turnstile>\<^sub>S\<^sub>5' (K i (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> K i p \<^bold>\<longrightarrow> K i q)\<close>
     using A2' .
   moreover have \<open>\<turnstile>\<^sub>S\<^sub>5' ((P \<^bold>\<longrightarrow> Q \<^bold>\<longrightarrow> R) \<^bold>\<longrightarrow> (Q \<^bold>\<and> P \<^bold>\<longrightarrow> R))\<close> for P Q R :: \<open>'i fm\<close>
     by (simp add: A1')
   ultimately show ?case
     using R1' by blast
 next
   case (Ax p)
   then show ?case
     using AT' S5'_B S5'_4
     by (metis Ax4.cases AxB.cases AxT.cases)
 qed (meson SystemS5'.intros)+
 
 lemma S5'_S5:
   fixes p :: \<open>('i :: countable) fm\<close>
   shows \<open>\<turnstile>\<^sub>S\<^sub>5' p \<Longrightarrow> \<turnstile>\<^sub>S\<^sub>5 p\<close>
 proof (induct p rule: SystemS5'.induct)
   case (AT' i p)
   then show ?case
     by (simp add: Ax AxT.intros)
 next
   case (A5' i p)
   then show ?case
     using completeness\<^sub>S\<^sub>5 neg_introspection by fast
 qed (meson AK.intros K_A2')+
 
 theorem main\<^sub>S\<^sub>5': \<open>valid\<^sub>S\<^sub>5 p \<longleftrightarrow> \<turnstile>\<^sub>S\<^sub>5' p\<close>
   using main\<^sub>S\<^sub>5 S5_S5' S5'_S5 by blast
 
 section \<open>Acknowledgements\<close>
 
 text \<open>
 The formalization is inspired by Berghofer's formalization of Henkin-style completeness.
 
   \<^item> Stefan Berghofer:
   First-Order Logic According to Fitting.
   \<^url>\<open>https://www.isa-afp.org/entries/FOL-Fitting.shtml\<close>
 \<close>
 
 end