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
@@ -1,1298 +1,1268 @@
 (*
   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 (\<open>\<^bold>\<bottom>\<close>)
   | Pro id
   | Dis \<open>'i fm\<close> \<open>'i fm\<close> (infixr \<open>\<^bold>\<or>\<close> 30)
   | Con \<open>'i fm\<close> \<open>'i fm\<close> (infixr \<open>\<^bold>\<and>\<close> 35)
   | Imp \<open>'i fm\<close> \<open>'i fm\<close> (infixr \<open>\<^bold>\<longrightarrow>\<close> 25)
   | K 'i \<open>'i fm\<close>
 
 abbreviation TT (\<open>\<^bold>\<top>\<close>) where
   \<open>TT \<equiv> \<^bold>\<bottom> \<^bold>\<longrightarrow> \<^bold>\<bottom>\<close>
 
 abbreviation Neg (\<open>\<^bold>\<not> _\<close> [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 (\<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> (\<open>_, _ \<Turnstile> _\<close> [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> \<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. \<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. \<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. \<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 refltrans :: \<open>('i, 'w) kripke \<Rightarrow> bool\<close> where
   \<open>refltrans M \<equiv> reflexive M \<and> transitive M\<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> \<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> \<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> \<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> \<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> \<open>w \<in> \<W> M\<close> unfolding reflexive_def by blast
   ultimately show \<open>M, w \<Turnstile> p\<close>
     using \<open>w \<in> \<W> M\<close> by simp
 qed
 
 theorem pos_introspection:
   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> \<W> M \<inter> \<K> M i w. M, v \<Turnstile> p\<close>
     by simp
   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> \<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> \<open>u \<in> \<W> M\<close>
     by auto
   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> \<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
 
 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> (\<open>_ \<turnstile> _\<close> [20, 20] 20)
   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>
 
+primrec imply :: \<open>'i fm list \<Rightarrow> 'i fm \<Rightarrow> 'i fm\<close> (infixr \<open>\<^bold>\<leadsto>\<close> 26) where
+  \<open>([] \<^bold>\<leadsto> q) = q\<close>
+| \<open>(p # ps \<^bold>\<leadsto> q) = (p \<^bold>\<longrightarrow> ps \<^bold>\<leadsto> q)\<close>
+
+abbreviation AK_assms (\<open>_, _ \<turnstile> _\<close> [20, 20, 20] 20) where
+  \<open>A, G \<turnstile> p \<equiv> \<exists>qs. set qs \<subseteq> G \<and> (A \<turnstile> qs \<^bold>\<leadsto> p)\<close>
+
 section \<open>Soundness\<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 W g r, w \<Turnstile> q) p\<close> for W g r
     by simp
   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 P :: \<open>('i, 'w) kripke \<Rightarrow> bool\<close>
   assumes \<open>\<And>M 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> (infixr \<open>\<^bold>\<leadsto>\<close> 26) where
-  \<open>([] \<^bold>\<leadsto> q) = q\<close>
-| \<open>(p # ps \<^bold>\<leadsto> q) = (p \<^bold>\<longrightarrow> ps \<^bold>\<leadsto> q)\<close>
-
 lemma K_imply_head: \<open>A \<turnstile> (p # ps \<^bold>\<leadsto> p)\<close>
 proof -
   have \<open>tautology (p # ps \<^bold>\<leadsto> p)\<close>
     by (induct ps) simp_all
   then show ?thesis
     using A1 by blast
 qed
 
 lemma K_imply_Cons:
   assumes \<open>A \<turnstile> ps \<^bold>\<leadsto> q\<close>
   shows \<open>A \<turnstile> p # ps \<^bold>\<leadsto> q\<close>
 proof -
   have \<open>A \<turnstile> (ps \<^bold>\<leadsto> q \<^bold>\<longrightarrow> p # ps \<^bold>\<leadsto> q)\<close>
     by (simp add: A1)
   with R1 assms show ?thesis .
 qed
 
 lemma K_right_mp:
   assumes \<open>A \<turnstile> ps \<^bold>\<leadsto> p\<close> \<open>A \<turnstile> ps \<^bold>\<leadsto> (p \<^bold>\<longrightarrow> q)\<close>
   shows \<open>A \<turnstile> ps \<^bold>\<leadsto> q\<close>
 proof -
   have \<open>tautology (ps \<^bold>\<leadsto> p \<^bold>\<longrightarrow> ps \<^bold>\<leadsto> (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> ps \<^bold>\<leadsto> q)\<close>
     by (induct ps) simp_all
   with A1 have \<open>A \<turnstile> ps \<^bold>\<leadsto> p \<^bold>\<longrightarrow> ps \<^bold>\<leadsto> (p \<^bold>\<longrightarrow> q) \<^bold>\<longrightarrow> ps \<^bold>\<leadsto> 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 (ps \<^bold>\<leadsto> r \<^bold>\<longrightarrow> qs \<^bold>\<leadsto> r)\<close>
 proof (rule ccontr)
   assume \<open>\<not> tautology (ps \<^bold>\<leadsto> r \<^bold>\<longrightarrow> qs \<^bold>\<leadsto> r)\<close>
   then obtain g h where \<open>\<not> eval g h (ps \<^bold>\<leadsto> r \<^bold>\<longrightarrow> qs \<^bold>\<leadsto> r)\<close>
     by blast
   then have \<open>eval g h (ps \<^bold>\<leadsto> r)\<close> \<open>\<not> eval g h (qs \<^bold>\<leadsto> 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 (qs \<^bold>\<leadsto> r)\<close>
       by (induct qs) simp_all
     then show ?thesis
       using \<open>\<not> eval g h (qs \<^bold>\<leadsto> r)\<close> by blast
   next
     case r
     then have \<open>eval g h (qs \<^bold>\<leadsto> r)\<close>
       by (induct qs) simp_all
     then show ?thesis
       using \<open>\<not> eval g h (qs \<^bold>\<leadsto> r)\<close> by blast
   qed
 qed
 
 lemma K_imply_weaken:
   assumes \<open>A \<turnstile> ps \<^bold>\<leadsto> q\<close> \<open>set ps \<subseteq> set ps'\<close>
   shows \<open>A \<turnstile> ps' \<^bold>\<leadsto> q\<close>
 proof -
   have \<open>tautology (ps \<^bold>\<leadsto> q \<^bold>\<longrightarrow> ps' \<^bold>\<leadsto> q)\<close>
     using \<open>set ps \<subseteq> set ps'\<close> tautology_imply_superset by blast
   then have \<open>A \<turnstile> (ps \<^bold>\<leadsto> q \<^bold>\<longrightarrow> ps' \<^bold>\<leadsto> q)\<close>
     using A1 by blast
   then show ?thesis
     using \<open>A \<turnstile> ps \<^bold>\<leadsto> q\<close> R1 by blast
 qed
 
 lemma imply_append: \<open>(ps @ ps' \<^bold>\<leadsto> q) = (ps \<^bold>\<leadsto> ps' \<^bold>\<leadsto> q)\<close>
   by (induct ps) simp_all
 
 lemma K_ImpI:
   assumes \<open>A \<turnstile> p # G \<^bold>\<leadsto> q\<close>
   shows \<open>A \<turnstile> G \<^bold>\<leadsto> (p \<^bold>\<longrightarrow> q)\<close>
 proof -
   have \<open>set (p # G) \<subseteq> set (G @ [p])\<close>
     by simp
   then have \<open>A \<turnstile> G @ [p] \<^bold>\<leadsto> q\<close>
     using assms K_imply_weaken by blast
   then have \<open>A \<turnstile> G \<^bold>\<leadsto> [p] \<^bold>\<leadsto> q\<close>
     using imply_append by metis
   then show ?thesis
     by simp
 qed
 
 lemma K_Boole:
   assumes \<open>A \<turnstile> (\<^bold>\<not> p) # G \<^bold>\<leadsto> \<^bold>\<bottom>\<close>
   shows \<open>A \<turnstile> G \<^bold>\<leadsto> p\<close>
 proof -
   have \<open>A \<turnstile> G \<^bold>\<leadsto> \<^bold>\<not> \<^bold>\<not> p\<close>
     using assms K_ImpI by blast
   moreover have \<open>tautology (G \<^bold>\<leadsto> \<^bold>\<not> \<^bold>\<not> p \<^bold>\<longrightarrow> G \<^bold>\<leadsto> p)\<close>
     by (induct G) simp_all
   then have \<open>A \<turnstile> (G \<^bold>\<leadsto> \<^bold>\<not> \<^bold>\<not> p \<^bold>\<longrightarrow> G \<^bold>\<leadsto> p)\<close>
     using A1 by blast
   ultimately show ?thesis
     using R1 by blast
 qed
 
 lemma K_DisE:
   assumes \<open>A \<turnstile> p # G \<^bold>\<leadsto> r\<close> \<open>A \<turnstile> q # G \<^bold>\<leadsto> r\<close> \<open>A \<turnstile> G \<^bold>\<leadsto> p \<^bold>\<or> q\<close>
   shows \<open>A \<turnstile> G \<^bold>\<leadsto> r\<close>
 proof -
   have \<open>tautology (p # G \<^bold>\<leadsto> r \<^bold>\<longrightarrow> q # G \<^bold>\<leadsto> r \<^bold>\<longrightarrow> G \<^bold>\<leadsto> p \<^bold>\<or> q \<^bold>\<longrightarrow> G \<^bold>\<leadsto> r)\<close>
     by (induct G) auto
   then have \<open>A \<turnstile> p # G \<^bold>\<leadsto> r \<^bold>\<longrightarrow> q # G \<^bold>\<leadsto> r \<^bold>\<longrightarrow> G \<^bold>\<leadsto> p \<^bold>\<or> q \<^bold>\<longrightarrow> G \<^bold>\<leadsto> r\<close>
     using A1 by blast
   then show ?thesis
     using assms R1 by blast
 qed
 
 lemma K_mp: \<open>A \<turnstile> p # (p \<^bold>\<longrightarrow> q) # G \<^bold>\<leadsto> q\<close>
   by (meson K_imply_head K_imply_weaken K_right_mp set_subset_Cons)
 
 lemma K_swap:
   assumes \<open>A \<turnstile> p # q # G \<^bold>\<leadsto> r\<close>
   shows \<open>A \<turnstile> q # p # G \<^bold>\<leadsto> r\<close>
   using assms K_ImpI by (metis imply.simps(1-2))
 
 lemma K_DisL:
   assumes \<open>A \<turnstile> p # ps \<^bold>\<leadsto> q\<close> \<open>A \<turnstile> p' # ps \<^bold>\<leadsto> q\<close>
   shows \<open>A \<turnstile> (p \<^bold>\<or> p') # ps \<^bold>\<leadsto> q\<close>
 proof -
   have \<open>A \<turnstile> p # (p \<^bold>\<or> p') # ps \<^bold>\<leadsto> q\<close> \<open>A \<turnstile> p' # (p \<^bold>\<or> p') # ps \<^bold>\<leadsto> q\<close>
     using assms K_swap K_imply_Cons by blast+
   moreover have \<open>A \<turnstile> (p \<^bold>\<or> p') # ps \<^bold>\<leadsto> 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 (G \<^bold>\<leadsto> q)\<close>
   shows \<open>A \<turnstile> map (K i) G \<^bold>\<leadsto> K i q\<close>
 proof -
   have \<open>A \<turnstile> (K i (G \<^bold>\<leadsto> q) \<^bold>\<longrightarrow> map (K i) G \<^bold>\<leadsto> 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 (a # G \<^bold>\<leadsto> q) \<^bold>\<longrightarrow> K i (G \<^bold>\<leadsto> q)\<close>
       by (simp add: A2)
     moreover have
       \<open>A \<turnstile> ((K i a \<^bold>\<and> K i (a # G \<^bold>\<leadsto> q) \<^bold>\<longrightarrow> K i (G \<^bold>\<leadsto> q)) \<^bold>\<longrightarrow>
         (K i (G \<^bold>\<leadsto> q) \<^bold>\<longrightarrow> map (K i) G \<^bold>\<leadsto> K i q) \<^bold>\<longrightarrow>
         (K i a \<^bold>\<and> K i (a # G \<^bold>\<leadsto> q) \<^bold>\<longrightarrow> map (K i) G \<^bold>\<leadsto> K i q))\<close>
       by (simp add: A1)
     ultimately have \<open>A \<turnstile> K i a \<^bold>\<and> K i (a # G \<^bold>\<leadsto> q) \<^bold>\<longrightarrow> map (K i) G \<^bold>\<leadsto> K i q\<close>
       using Cons R1 by blast
     moreover have
       \<open>A \<turnstile> ((K i a \<^bold>\<and> K i (a # G \<^bold>\<leadsto> q) \<^bold>\<longrightarrow> map (K i) G \<^bold>\<leadsto> K i q) \<^bold>\<longrightarrow>
         (K i (a # G \<^bold>\<leadsto> q) \<^bold>\<longrightarrow> K i a \<^bold>\<longrightarrow> map (K i) G \<^bold>\<leadsto> K i q))\<close>
       by (simp add: A1)
     ultimately have \<open>A \<turnstile> (K i (a # G \<^bold>\<leadsto> q) \<^bold>\<longrightarrow> K i a \<^bold>\<longrightarrow> map (K i) G \<^bold>\<leadsto> 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>Strong Soundness\<close>
+
+abbreviation validStar :: \<open>(('i, 'w) kripke \<Rightarrow> bool) \<Rightarrow> 'i fm set \<Rightarrow> 'i fm \<Rightarrow> bool\<close> (\<open>valid\<star>\<close>) where
+  \<open>valid\<star> P G p \<equiv> \<forall>M. \<forall>w \<in> \<W> M. P M \<longrightarrow> (\<forall>q \<in> G. M, w \<Turnstile> q) \<longrightarrow> M, w \<Turnstile> p\<close>
+
+corollary soundness_imply:
+  assumes \<open>\<And>M w p. A p \<Longrightarrow> P M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
+  shows \<open>A \<turnstile> ps \<^bold>\<leadsto> p \<Longrightarrow> valid\<star> P (set ps) p\<close>
+proof (induct ps arbitrary: p)
+  case Nil
+  then show ?case
+    using soundness[of A P p] assms by simp
+next
+  case (Cons a ps)
+  then show ?case
+    using K_ImpI by fastforce
+qed
+
+theorem strong_soundness:
+  assumes \<open>\<And>M w p. A p \<Longrightarrow> P M \<Longrightarrow> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
+  shows \<open>A, G \<turnstile> p \<Longrightarrow> valid\<star> P G p\<close>
+proof safe
+  fix qs w and M :: \<open>('a, 'b) kripke\<close>
+  assume \<open>A \<turnstile> qs \<^bold>\<leadsto> p\<close>
+  moreover assume \<open>set qs \<subseteq> G\<close> \<open>\<forall>q \<in> G. M, w \<Turnstile> q\<close>
+  then have \<open>\<forall>q \<in> set qs. M, w \<Turnstile> q\<close>
+    using \<open>set qs \<subseteq> G\<close> by blast
+  moreover assume \<open>P M\<close> \<open>w \<in> \<W> M\<close>
+  ultimately show \<open>M, w \<Turnstile> p\<close>
+    using soundness_imply[of A P qs p] assms 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> S' \<^bold>\<leadsto> \<^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> p # V' \<^bold>\<leadsto> \<^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> V' \<^bold>\<leadsto> \<^bold>\<bottom>\<close>
     using assms unfolding consistent_def by blast
   then have *: \<open>A \<turnstile> p # V' \<^bold>\<leadsto> \<^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> p # ?S \<^bold>\<leadsto> \<^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_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> p # V' \<^bold>\<leadsto> \<^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> q # V' \<^bold>\<leadsto> \<^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> p # S' \<^bold>\<leadsto> \<^bold>\<bottom>\<close> and
     T': \<open>set T' \<subseteq> V\<close> \<open>A \<turnstile> q # T' \<^bold>\<leadsto> \<^bold>\<bottom>\<close>
     using \<open>consistent A V\<close> inconsistent_subset by metis
 
   from S' have p: \<open>A \<turnstile> p # S' @ T' \<^bold>\<leadsto> \<^bold>\<bottom>\<close>
     by (metis K_imply_weaken Un_upper1 append_Cons set_append)
   moreover from T' have q: \<open>A \<turnstile> q # S' @ T' \<^bold>\<leadsto> \<^bold>\<bottom>\<close>
     by (metis K_imply_head K_right_mp R1 imply.simps(2) imply_append)
   ultimately have \<open>A \<turnstile> (p \<^bold>\<or> q) # S' @ T' \<^bold>\<leadsto> \<^bold>\<bottom>\<close>
     using K_DisL by blast
   then have \<open>A \<turnstile> S' @ T' \<^bold>\<leadsto> \<^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> W' \<^bold>\<leadsto> \<^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> G \<^bold>\<leadsto> p\<close>
   using assms K_Boole K_imply_weaken unfolding consistent_def
   by (metis insert_is_Un list.simps(15))
 
 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> p # (p \<^bold>\<longrightarrow> q) # V' \<^bold>\<leadsto> \<^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> q # V' \<^bold>\<leadsto> \<^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> 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> S' \<^bold>\<leadsto> \<^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> S' \<^bold>\<leadsto> \<^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
 
 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}\<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>
 
 abbreviation canonical :: \<open>('i fm \<Rightarrow> bool) \<Rightarrow> ('i, 'i fm set) kripke\<close> where
   \<open>canonical A \<equiv> Kripke (mcss A) pi (reach A)\<close>
 
 lemma truth_lemma:
   fixes A and p :: \<open>('i :: countable) fm\<close>
   assumes \<open>consistent A V\<close> and \<open>maximal A V\<close>
   shows \<open>p \<in> V \<longleftrightarrow> canonical A, V \<Turnstile> p\<close>
   using assms
 proof (induct p arbitrary: V)
   case FF
   then show ?case
   proof safe
     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>canonical A, V \<Turnstile> \<^bold>\<bottom>\<close> ..
   next
     assume \<open>canonical A, V \<Turnstile> \<^bold>\<bottom>\<close>
     then show \<open>\<^bold>\<bottom> \<in> V\<close>
       by simp
   qed
 next
   case (Pro x)
   then show ?case
     by simp
 next
   case (Dis p q)
   then show ?case
   proof safe
     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>canonical A, V \<Turnstile> (p \<^bold>\<or> q)\<close>
       using Dis by simp
   next
     assume \<open>canonical A, V \<Turnstile> (p \<^bold>\<or> q)\<close>
     then consider \<open>canonical A, V \<Turnstile> p\<close> | \<open>canonical A, V \<Turnstile> q\<close>
       by auto
     then have \<open>p \<in> V \<or> q \<in> V\<close>
       using Dis by auto
     moreover have \<open>A \<turnstile> p \<^bold>\<longrightarrow> p \<^bold>\<or> q\<close> \<open>A \<turnstile> q \<^bold>\<longrightarrow> p \<^bold>\<or> q\<close>
       by (auto simp: A1)
     ultimately show \<open>(p \<^bold>\<or> q) \<in> V\<close>
       using Dis.prems deriv_in_maximal consequent_in_maximal by blast
   qed
 next
   case (Con p q)
   then show ?case
   proof safe
     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>canonical A, V \<Turnstile> (p \<^bold>\<and> q)\<close>
       using Con by simp
   next
     assume \<open>canonical A, V \<Turnstile> (p \<^bold>\<and> q)\<close>
     then have \<open>canonical A, V \<Turnstile> p\<close> \<open>canonical A, V \<Turnstile> q\<close>
       by auto
     then have \<open>p \<in> V\<close> \<open>q \<in> V\<close>
       using Con by auto
     moreover have \<open>A \<turnstile> p \<^bold>\<longrightarrow> q \<^bold>\<longrightarrow> p \<^bold>\<and> q\<close>
       by (auto simp: A1)
     ultimately show \<open>(p \<^bold>\<and> q) \<in> V\<close>
       using Con.prems deriv_in_maximal consequent_in_maximal by blast
   qed
 next
   case (Imp p q)
   then show ?case
   proof safe
     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 have \<open>p \<notin> V \<or> q \<in> V\<close>
       using Imp.prems exactly_one_in_maximal by blast
     then show \<open>canonical A, V \<Turnstile> (p \<^bold>\<longrightarrow> q)\<close>
       using Imp by simp
   next
     assume \<open>canonical A, V \<Turnstile> (p \<^bold>\<longrightarrow> q)\<close>
     then consider \<open>\<not> canonical A, V \<Turnstile> p\<close> | \<open>canonical A, V \<Turnstile> q\<close>
       by auto
     then have \<open>p \<notin> V \<or> q \<in> V\<close>
       using Imp by auto
     then have \<open>(\<^bold>\<not> p) \<in> V \<or> q \<in> V\<close>
       using Imp.prems exactly_one_in_maximal by blast
     moreover have \<open>A \<turnstile> \<^bold>\<not> p \<^bold>\<longrightarrow> p \<^bold>\<longrightarrow> q\<close> \<open>A \<turnstile> q \<^bold>\<longrightarrow> p \<^bold>\<longrightarrow> q\<close>
       by (auto simp: A1)
     ultimately show \<open>(p \<^bold>\<longrightarrow> q) \<in> V\<close>
       using Imp.prems deriv_in_maximal consequent_in_maximal by blast
   qed
 next
   case (K i p)
   then show ?case
   proof safe
     assume \<open>K i p \<in> V\<close>
     then show \<open>canonical A, V \<Turnstile> K i p\<close>
       using K.hyps by auto
   next
     assume \<open>canonical 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>canonical A, W \<Turnstile> \<^bold>\<not> p\<close>
         using K \<open>consistent A V\<close> exactly_one_in_maximal by auto
       moreover have \<open>W \<in> reach A i V\<close> \<open>W \<in> mcss A\<close>
         using W by simp_all
       ultimately have \<open>canonical A, V \<Turnstile> \<^bold>\<not> K i p\<close>
         by auto
       then show False
         using \<open>canonical 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> L \<^bold>\<leadsto> p\<close>
       using W(4) inconsistent_imply by blast
     then have \<open>A \<turnstile> K i (L \<^bold>\<leadsto> p)\<close>
       using R2 by fast
     then have \<open>A \<turnstile> map (K i) L \<^bold>\<leadsto> K i p\<close>
       using K_distrib_K_imp by fast
     then have \<open>(map (K i) L \<^bold>\<leadsto> 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>(map (K i) L \<^bold>\<leadsto> 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
 qed
 
 lemma canonical_model:
   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> canonical 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>
 
 abbreviation valid :: \<open>(('i :: countable, 'i fm set) kripke \<Rightarrow> bool) \<Rightarrow> 'i fm set \<Rightarrow> 'i fm \<Rightarrow> bool\<close>
-  where \<open>valid P G p \<equiv> \<forall>M. \<forall>w \<in> \<W> M. P M \<longrightarrow> (\<forall>q \<in> G. M, w \<Turnstile> q) \<longrightarrow> M, w \<Turnstile> p\<close>
+  (\<open>valid\<close>) where \<open>valid \<equiv> valid\<star>\<close>
 
 theorem strong_completeness:
   assumes \<open>valid P G p\<close> and \<open>P (canonical A)\<close>
   shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (A \<turnstile> qs \<^bold>\<leadsto> p)\<close>
 proof (rule ccontr)
   assume \<open>\<nexists>qs. set qs \<subseteq> G \<and> (A \<turnstile> qs \<^bold>\<leadsto> p)\<close>
   then have *: \<open>\<forall>qs. set qs \<subseteq> G \<longrightarrow> \<not> (A \<turnstile> (\<^bold>\<not> p) # qs \<^bold>\<leadsto> \<^bold>\<bottom>)\<close>
     using K_Boole by blast
 
   let ?S = \<open>{\<^bold>\<not> p} \<union> G\<close>
   let ?V = \<open>Extend A ?S from_nat\<close>
   let ?M = \<open>canonical A\<close>
 
   have \<open>consistent A ?S\<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>
     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 assms by simp
   then show False
     using \<open>?M, ?V \<Turnstile> (\<^bold>\<not> p)\<close> by simp
 qed
 
 corollary completeness:
   assumes \<open>valid P {} p\<close> and \<open>P (canonical A)\<close>
   shows \<open>A \<turnstile> p\<close>
   using assms strong_completeness[where G=\<open>{}\<close>] by simp
 
 corollary completeness\<^sub>A:
   assumes \<open>valid (\<lambda>_. True) {} p\<close>
   shows \<open>A \<turnstile> p\<close>
   using assms completeness by blast
 
 section \<open>System K\<close>
 
-abbreviation SystemK :: \<open>'i fm \<Rightarrow> bool\<close> (\<open>\<turnstile>\<^sub>K _\<close> [20] 20) where
-  \<open>\<turnstile>\<^sub>K p \<equiv> (\<lambda>_. False) \<turnstile> p\<close>
+abbreviation SystemK (\<open>_ \<turnstile>\<^sub>K _\<close> [50] 50) where
+  \<open>G \<turnstile>\<^sub>K p \<equiv> (\<lambda>_. False), G \<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
+lemma strong_soundness\<^sub>K: \<open>G \<turnstile>\<^sub>K p \<Longrightarrow> valid\<star> P G p\<close>
+  using strong_soundness[of \<open>\<lambda>_. False\<close> \<open>\<lambda>_. True\<close>] by fast
 
 abbreviation \<open>valid\<^sub>K \<equiv> valid (\<lambda>_. True)\<close>
 
-theorem strong_completeness\<^sub>K:
-  assumes \<open>valid\<^sub>K G p\<close>
-  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (\<turnstile>\<^sub>K qs \<^bold>\<leadsto> p)\<close>
-  using assms strong_completeness[where P=\<open>\<lambda>_. True\<close>] by blast
-
-corollary completeness\<^sub>K:
-  assumes \<open>valid\<^sub>K {} p\<close>
-  shows \<open>\<turnstile>\<^sub>K p\<close>
-  using completeness\<^sub>A assms .
+lemma strong_completeness\<^sub>K: \<open>valid\<^sub>K G p \<Longrightarrow> G \<turnstile>\<^sub>K p\<close>
+  using strong_completeness[of \<open>\<lambda>_. True\<close>] by blast
 
-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 fast
-qed
+theorem main\<^sub>K: \<open>valid\<^sub>K G p \<longleftrightarrow> G \<turnstile>\<^sub>K p\<close>
+  using strong_soundness\<^sub>K[of G p] strong_completeness\<^sub>K[of G p] by fast
 
-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
+corollary \<open>valid\<^sub>K G p \<Longrightarrow> valid\<star> (\<lambda>_. True) G p\<close>
+  using strong_soundness\<^sub>K[of G p] strong_completeness\<^sub>K[of G p] by fast
 
 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> (\<open>\<turnstile>\<^sub>T _\<close> [20] 20) where
-  \<open>\<turnstile>\<^sub>T p \<equiv> AxT \<turnstile> p\<close>
+abbreviation SystemT (\<open>_ \<turnstile>\<^sub>T _\<close> [50, 50] 50) where
+  \<open>G \<turnstile>\<^sub>T p \<equiv> AxT, G \<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> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
-  using soundness soundness_AxT .
+lemma strong_soundness\<^sub>T: \<open>G \<turnstile>\<^sub>T p \<Longrightarrow> valid\<star> reflexive G p\<close>
+  using strong_soundness soundness_AxT .
 
 lemma AxT_reflexive:
   assumes \<open>AxT \<le> A\<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 fast
   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 reflexive\<^sub>T:
   assumes \<open>AxT \<le> A\<close>
   shows \<open>reflexive (canonical A)\<close>
   unfolding reflexive_def
 proof safe
   fix i V
   assume \<open>V \<in> \<W> (canonical 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> (canonical A) i V\<close>
     by simp
 qed
 
 abbreviation \<open>valid\<^sub>T \<equiv> valid reflexive\<close>
 
-theorem strong_completeness\<^sub>T:
-  assumes \<open>valid\<^sub>T G p\<close>
-  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (\<turnstile>\<^sub>T qs \<^bold>\<leadsto> p)\<close>
-  using assms strong_completeness reflexive\<^sub>T by blast
+lemma strong_completeness\<^sub>T: \<open>valid\<^sub>T G p \<Longrightarrow> G \<turnstile>\<^sub>T p\<close>
+  using strong_completeness reflexive\<^sub>T by blast
 
-corollary completeness\<^sub>T:
-  assumes \<open>valid\<^sub>T {} p\<close>
-  shows \<open>\<turnstile>\<^sub>T p\<close>
-  using assms completeness reflexive\<^sub>T by blast
+theorem main\<^sub>T: \<open>valid\<^sub>T G p \<longleftrightarrow> G \<turnstile>\<^sub>T p\<close>
+  using strong_soundness\<^sub>T[of G p] strong_completeness\<^sub>T[of G p] by fast
 
-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> \<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
+corollary \<open>valid\<^sub>T G p \<longrightarrow> valid\<star> reflexive G p\<close>
+  using strong_soundness\<^sub>T[of G p] strong_completeness\<^sub>T[of G p] 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> (\<open>\<turnstile>\<^sub>K\<^sub>B _\<close> [20] 20) where
-  \<open>\<turnstile>\<^sub>K\<^sub>B p \<equiv> AxB \<turnstile> p\<close>
+abbreviation SystemKB (\<open>_ \<turnstile>\<^sub>K\<^sub>B _\<close> [50, 50] 50) where
+  \<open>G \<turnstile>\<^sub>K\<^sub>B p \<equiv> AxB, G \<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> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
-  using soundness soundness_AxB .
+lemma strong_soundness\<^sub>K\<^sub>B: \<open>G \<turnstile>\<^sub>K\<^sub>B p \<Longrightarrow> valid\<star> symmetric G p\<close>
+  using strong_soundness soundness_AxB .
 
 lemma AxB_symmetric':
   assumes \<open>AxB \<le> A\<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 symmetric\<^sub>K\<^sub>B:
   assumes \<open>AxB \<le> A\<close>
   shows \<open>symmetric (canonical A)\<close>
   unfolding symmetric_def
 proof (intro allI ballI)
   fix i V W
   assume \<open>V \<in> \<W> (canonical A)\<close> \<open>W \<in> \<W> (canonical 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>
     by metis
-  then show
-    \<open>(W \<in> \<K> (canonical A) i V) = (V \<in> \<K> (canonical A) i W)\<close>
+  then show \<open>(W \<in> \<K> (canonical A) i V) = (V \<in> \<K> (canonical A) i W)\<close>
     by simp
 qed
 
 abbreviation \<open>valid\<^sub>K\<^sub>B \<equiv> valid symmetric\<close>
 
-theorem strong_completeness\<^sub>K\<^sub>B:
-  assumes \<open>valid\<^sub>K\<^sub>B G p\<close>
-  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (\<turnstile>\<^sub>K\<^sub>B qs \<^bold>\<leadsto> p)\<close>
-  using assms strong_completeness symmetric\<^sub>K\<^sub>B by blast
+lemma strong_completeness\<^sub>K\<^sub>B: \<open>valid\<^sub>K\<^sub>B G p \<Longrightarrow> G \<turnstile>\<^sub>K\<^sub>B p\<close>
+  using strong_completeness symmetric\<^sub>K\<^sub>B by blast
 
-corollary completeness\<^sub>K\<^sub>B:
-  assumes \<open>valid\<^sub>K\<^sub>B {} p\<close>
-  shows \<open>\<turnstile>\<^sub>K\<^sub>B p\<close>
-  using assms strong_completeness\<^sub>K\<^sub>B[where G=\<open>{}\<close>] by auto
+theorem main\<^sub>K\<^sub>B: \<open>valid\<^sub>K\<^sub>B G p \<longleftrightarrow> G \<turnstile>\<^sub>K\<^sub>B p\<close>
+  using strong_soundness\<^sub>K\<^sub>B[of G p] strong_completeness\<^sub>K\<^sub>B[of G p] by fast
 
-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> \<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
+corollary \<open>valid\<^sub>K\<^sub>B G p \<longrightarrow> valid\<star> symmetric G p\<close>
+  using strong_soundness\<^sub>K\<^sub>B[of G p] strong_completeness\<^sub>K\<^sub>B[of G p] 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> (\<open>\<turnstile>\<^sub>K\<^sub>4 _\<close> [20] 20) where
-  \<open>\<turnstile>\<^sub>K\<^sub>4 p \<equiv> Ax4 \<turnstile> p\<close>
+abbreviation SystemK4 (\<open>_ \<turnstile>\<^sub>K\<^sub>4 _\<close> [50, 50] 50) where
+  \<open>G \<turnstile>\<^sub>K\<^sub>4 p \<equiv> Ax4, G \<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> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
-  using soundness soundness_Ax4 .
+lemma strong_soundness\<^sub>K\<^sub>4: \<open>G \<turnstile>\<^sub>K\<^sub>4 p \<Longrightarrow> valid\<star> transitive G p\<close>
+  using strong_soundness soundness_Ax4 .
 
 lemma Ax4_transitive:
   assumes \<open>Ax4 \<le> A\<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 fast
   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 transitive\<^sub>K\<^sub>4:
   assumes \<open>Ax4 \<le> A\<close>
   shows \<open>transitive (canonical A)\<close>
   unfolding transitive_def
 proof safe
   fix i U V W
   assume \<open>V \<in> \<W> (canonical A)\<close>
   then have \<open>consistent A V\<close> \<open>maximal A V\<close>
     by simp_all
   moreover assume
     \<open>W \<in> \<K> (canonical A) i V\<close>
     \<open>U \<in> \<K> (canonical A) i W\<close>
   ultimately have \<open>U \<in> reach A i V\<close>
     using Ax4_transitive assms by simp
   then show \<open>U \<in> \<K> (canonical A) i V\<close>
     by simp
 qed
 
 abbreviation \<open>valid\<^sub>K\<^sub>4 \<equiv> valid transitive\<close>
 
-theorem strong_completeness\<^sub>K\<^sub>4:
-  assumes \<open>valid\<^sub>K\<^sub>4 G p\<close>
-  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (\<turnstile>\<^sub>K\<^sub>4 qs \<^bold>\<leadsto> p)\<close>
-  using assms strong_completeness transitive\<^sub>K\<^sub>4 by blast
+lemma strong_completeness\<^sub>K\<^sub>4: \<open>valid\<^sub>K\<^sub>4 G p \<Longrightarrow> G \<turnstile>\<^sub>K\<^sub>4 p\<close>
+  using strong_completeness transitive\<^sub>K\<^sub>4 by blast
 
-corollary completeness\<^sub>K\<^sub>4:
-  assumes \<open>valid\<^sub>K\<^sub>4 {} p\<close>
-  shows \<open>\<turnstile>\<^sub>K\<^sub>4 p\<close>
-  using assms strong_completeness\<^sub>K\<^sub>4[where G=\<open>{}\<close>] by auto
+theorem main\<^sub>K\<^sub>4: \<open>valid\<^sub>K\<^sub>4 G p \<longleftrightarrow> G \<turnstile>\<^sub>K\<^sub>4 p\<close>
+  using strong_soundness\<^sub>K\<^sub>4[of G p] strong_completeness\<^sub>K\<^sub>4[of G p] by fast
 
-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> \<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
+corollary \<open>valid\<^sub>K\<^sub>4 G p \<longrightarrow> valid\<star> transitive G p\<close>
+  using strong_soundness\<^sub>K\<^sub>4[of G p] strong_completeness\<^sub>K\<^sub>4[of G p] 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> (\<open>\<turnstile>\<^sub>S\<^sub>4 _\<close> [20] 20) where
-  \<open>\<turnstile>\<^sub>S\<^sub>4 p \<equiv> AxT \<oplus> Ax4 \<turnstile> p\<close>
+abbreviation SystemS4 (\<open>_ \<turnstile>\<^sub>S\<^sub>4 _\<close> [50, 50] 50) where
+  \<open>G \<turnstile>\<^sub>S\<^sub>4 p \<equiv> AxT \<oplus> Ax4, G \<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> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
-  using soundness soundness_AxT4 .
+lemma strong_soundness\<^sub>S\<^sub>4: \<open>G \<turnstile>\<^sub>S\<^sub>4 p \<Longrightarrow> valid\<star> refltrans G p\<close>
+  using strong_soundness soundness_AxT4 .
 
 abbreviation \<open>valid\<^sub>S\<^sub>4 \<equiv> valid refltrans\<close>
 
-theorem strong_completeness\<^sub>S\<^sub>4:
-  assumes \<open>valid\<^sub>S\<^sub>4 G p\<close>
-  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (\<turnstile>\<^sub>S\<^sub>4 qs \<^bold>\<leadsto> p)\<close>
-  using assms strong_completeness[where P=refltrans]
-    reflexive\<^sub>T[where A=\<open>AxT \<oplus> Ax4\<close>] transitive\<^sub>K\<^sub>4[where A=\<open>AxT \<oplus> Ax4\<close>] by blast
+lemma strong_completeness\<^sub>S\<^sub>4: \<open>valid\<^sub>S\<^sub>4 G p \<Longrightarrow> G \<turnstile>\<^sub>S\<^sub>4 p\<close>
+  using strong_completeness[of refltrans] reflexive\<^sub>T[of \<open>AxT \<oplus> Ax4\<close>] transitive\<^sub>K\<^sub>4[of \<open>AxT \<oplus> Ax4\<close>]
+  by blast
 
-corollary completeness\<^sub>S\<^sub>4:
-  assumes \<open>valid\<^sub>S\<^sub>4 {} p\<close>
-  shows \<open>\<turnstile>\<^sub>S\<^sub>4 p\<close>
-  using assms strong_completeness\<^sub>S\<^sub>4[where G=\<open>{}\<close>] by auto
+theorem main\<^sub>S\<^sub>4: \<open>valid\<^sub>S\<^sub>4 G p \<longleftrightarrow> G \<turnstile>\<^sub>S\<^sub>4 p\<close>
+  using strong_soundness\<^sub>S\<^sub>4[of G p] strong_completeness\<^sub>S\<^sub>4[of G p] by fast
 
-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> \<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
+corollary \<open>valid\<^sub>S\<^sub>4 G p \<longrightarrow> valid\<star> refltrans G p\<close>
+  using strong_soundness\<^sub>S\<^sub>4[of G p] strong_completeness\<^sub>S\<^sub>4[of G p] by fast
 
 section \<open>System S5\<close>
 
-abbreviation SystemS5 :: \<open>'i fm \<Rightarrow> bool\<close> (\<open>\<turnstile>\<^sub>S\<^sub>5 _\<close> [20] 20) where
-  \<open>\<turnstile>\<^sub>S\<^sub>5 p \<equiv> AxT \<oplus> AxB \<oplus> Ax4 \<turnstile> p\<close>
+abbreviation SystemS5 (\<open>_ \<turnstile>\<^sub>S\<^sub>5 _\<close> [50, 50] 50) where
+  \<open>G \<turnstile>\<^sub>S\<^sub>5 p \<equiv> AxT \<oplus> AxB \<oplus> Ax4, G \<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> 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> w \<in> \<W> M \<Longrightarrow> M, w \<Turnstile> p\<close>
-  using soundness soundness_AxTB4 .
+lemma strong_soundness\<^sub>S\<^sub>5: \<open>G \<turnstile>\<^sub>S\<^sub>5 p \<Longrightarrow> valid\<star> equivalence G p\<close>
+  using strong_soundness soundness_AxTB4 .
 
 abbreviation \<open>valid\<^sub>S\<^sub>5 \<equiv> valid equivalence\<close>
 
-theorem strong_completeness\<^sub>S\<^sub>5:
-  assumes \<open>valid\<^sub>S\<^sub>5 G p\<close>
-  shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (\<turnstile>\<^sub>S\<^sub>5 qs \<^bold>\<leadsto> p)\<close>
-  using assms strong_completeness[where P=equivalence]
-    reflexive\<^sub>T[where A=AxTB4] symmetric\<^sub>K\<^sub>B[where A=AxTB4] transitive\<^sub>K\<^sub>4[where A=AxTB4]
+lemma strong_completeness\<^sub>S\<^sub>5: \<open>valid\<^sub>S\<^sub>5 G p \<Longrightarrow> G \<turnstile>\<^sub>S\<^sub>5 p\<close>
+  using strong_completeness[of equivalence]
+    reflexive\<^sub>T[of AxTB4] symmetric\<^sub>K\<^sub>B[of AxTB4] transitive\<^sub>K\<^sub>4[of AxTB4]
   by blast
 
-corollary completeness\<^sub>S\<^sub>5:
-  assumes \<open>valid\<^sub>S\<^sub>5 {} p\<close>
-  shows \<open>\<turnstile>\<^sub>S\<^sub>5 p\<close>
-  using assms strong_completeness\<^sub>S\<^sub>5[where G=\<open>{}\<close>] by auto
+theorem main\<^sub>S\<^sub>5: \<open>valid\<^sub>S\<^sub>5 G p \<longleftrightarrow> G \<turnstile>\<^sub>S\<^sub>5 p\<close>
+  using strong_soundness\<^sub>S\<^sub>5[of G p] strong_completeness\<^sub>S\<^sub>5[of G p] by fast
 
-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> \<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
+corollary \<open>valid\<^sub>S\<^sub>5 G p \<longrightarrow> valid\<star> equivalence G p\<close>
+  using strong_soundness\<^sub>S\<^sub>5[of G p] strong_completeness\<^sub>S\<^sub>5[of G p] by fast
 
 subsection \<open>Traditional formulation\<close>
 
 inductive SystemS5' :: \<open>'i fm \<Rightarrow> bool\<close> (\<open>\<turnstile>\<^sub>S\<^sub>5'' _\<close> [20] 20) 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'_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'_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>
+lemma S5_S5': \<open>AxTB4 \<turnstile> 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>
+  shows \<open>\<turnstile>\<^sub>S\<^sub>5' p \<Longrightarrow> AxTB4 \<turnstile> 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)
+  let ?p = \<open>\<^bold>\<not> K i p \<^bold>\<longrightarrow> K i (\<^bold>\<not> K i p)\<close>
+  have \<open>valid\<^sub>S\<^sub>5 {} ?p\<close>
+    by (meson neg_introspection)
   then show ?case
-    using completeness\<^sub>S\<^sub>5 neg_introspection by fast
+    using strong_completeness\<^sub>S\<^sub>5[of \<open>{}\<close> \<open>?p\<close>] by fastforce 
 qed (meson AK.intros K_A2')+
 
 theorem strong_completeness\<^sub>S\<^sub>5':
   assumes \<open>valid\<^sub>S\<^sub>5 G p\<close>
   shows \<open>\<exists>qs. set qs \<subseteq> G \<and> (\<turnstile>\<^sub>S\<^sub>5' qs \<^bold>\<leadsto> p)\<close>
   using assms strong_completeness\<^sub>S\<^sub>5 S5_S5' S5'_S5 by blast
 
 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
+  using main\<^sub>S\<^sub>5[of \<open>{}\<close> p] S5_S5' S5'_S5 by auto
 
 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