# HG changeset patch
# User wenzelm
# Date 1452638433 -3600
# Node ID 7355fd313cf85ce8a1eb59e9ee43750cc34d0341
# Parent  ec2f0dad8b98b98ddaf77e0b62d9d8e3755f157f
misc tuning and modernization;

diff -r ec2f0dad8b98 -r 7355fd313cf8 src/HOL/HOLCF/IOA/CompoTraces.thy
--- a/src/HOL/HOLCF/IOA/CompoTraces.thy	Tue Jan 12 20:05:53 2016 +0100
+++ b/src/HOL/HOLCF/IOA/CompoTraces.thy	Tue Jan 12 23:40:33 2016 +0100
@@ -8,58 +8,47 @@
 imports CompoScheds ShortExecutions
 begin
 
-definition mksch :: "('a,'s)ioa => ('a,'t)ioa => 'a Seq -> 'a Seq -> 'a Seq -> 'a Seq"
-where
-  "mksch A B = (fix$(LAM h tr schA schB. case tr of
-       nil => nil
-    | x##xs =>
-      (case x of
-        UU => UU
-      | Def y =>
-         (if y:act A then
-             (if y:act B then
-                   ((Takewhile (%a. a:int A)$schA)
-                      @@ (Takewhile (%a. a:int B)$schB)
-                           @@ (y\<leadsto>(h$xs
-                                    $(TL$(Dropwhile (%a. a:int A)$schA))
-                                    $(TL$(Dropwhile (%a. a:int B)$schB))
-                    )))
-              else
-                 ((Takewhile (%a. a:int A)$schA)
-                  @@ (y\<leadsto>(h$xs
-                           $(TL$(Dropwhile (%a. a:int A)$schA))
-                           $schB)))
-              )
-          else
-             (if y:act B then
-                 ((Takewhile (%a. a:int B)$schB)
-                     @@ (y\<leadsto>(h$xs
-                              $schA
-                              $(TL$(Dropwhile (%a. a:int B)$schB))
-                              )))
-             else
-               UU
-             )
-         )
-       )))"
+definition mksch ::
+    "('a, 's) ioa \<Rightarrow> ('a, 't) ioa \<Rightarrow> 'a Seq \<rightarrow> 'a Seq \<rightarrow> 'a Seq \<rightarrow> 'a Seq"
+  where "mksch A B =
+    (fix $
+      (LAM h tr schA schB.
+        case tr of
+          nil \<Rightarrow> nil
+        | x ## xs \<Rightarrow>
+            (case x of
+              UU \<Rightarrow> UU
+            | Def y \<Rightarrow>
+                (if y \<in> act A then
+                  (if y \<in> act B then
+                    ((Takewhile (\<lambda>a. a \<in> int A) $ schA) @@
+                     (Takewhile (\<lambda>a. a \<in> int B) $ schB) @@
+                     (y \<leadsto> (h $ xs $ (TL $ (Dropwhile (\<lambda>a. a \<in> int A) $ schA))
+                                   $ (TL $ (Dropwhile (\<lambda>a. a \<in> int B) $ schB)))))
+                   else
+                    ((Takewhile (\<lambda>a. a \<in> int A) $ schA) @@
+                     (y \<leadsto> (h $ xs $ (TL $ (Dropwhile (\<lambda>a. a \<in> int A) $ schA)) $ schB))))
+                 else
+                  (if y \<in> act B then
+                    ((Takewhile (\<lambda>a. a \<in> int B) $ schB) @@
+                     (y \<leadsto> (h $ xs $ schA $ (TL $ (Dropwhile (\<lambda>a. a \<in> int B) $ schB)))))
+                   else UU)))))"
 
-definition par_traces ::"['a trace_module,'a trace_module] => 'a trace_module"
-where
-  "par_traces TracesA TracesB =
-      (let trA = fst TracesA; sigA = snd TracesA;
-           trB = fst TracesB; sigB = snd TracesB
-       in
-       (    {tr. Filter (%a. a:actions sigA)$tr : trA}
-        Int {tr. Filter (%a. a:actions sigB)$tr : trB}
-        Int {tr. Forall (%x. x:(externals sigA Un externals sigB)) tr},
+definition par_traces :: "'a trace_module \<Rightarrow> 'a trace_module \<Rightarrow> 'a trace_module"
+  where "par_traces TracesA TracesB =
+    (let
+      trA = fst TracesA; sigA = snd TracesA;
+      trB = fst TracesB; sigB = snd TracesB
+     in
+       ({tr. Filter (\<lambda>a. a \<in> actions sigA) $ tr \<in> trA} \<inter>
+        {tr. Filter (\<lambda>a. a \<in> actions sigB) $ tr \<in> trB} \<inter>
+        {tr. Forall (\<lambda>x. x \<in> externals sigA \<union> externals sigB) tr},
         asig_comp sigA sigB))"
 
-axiomatization
-where
-  finiteR_mksch:
-    "Finite (mksch A B$tr$x$y) \<Longrightarrow> Finite tr"
+axiomatization where
+  finiteR_mksch: "Finite (mksch A B $ tr $ x $ y) \<Longrightarrow> Finite tr"
 
-lemma finiteR_mksch': "\<not> Finite tr \<Longrightarrow> \<not> Finite (mksch A B$tr$x$y)"
+lemma finiteR_mksch': "\<not> Finite tr \<Longrightarrow> \<not> Finite (mksch A B $ tr $ x $ y)"
   by (blast intro: finiteR_mksch)
 
 
@@ -69,89 +58,79 @@
 subsection "mksch rewrite rules"
 
 lemma mksch_unfold:
-"mksch A B = (LAM tr schA schB. case tr of
-       nil => nil
-    | x##xs =>
-      (case x of
-        UU => UU
-      | Def y =>
-         (if y:act A then
-             (if y:act B then
-                   ((Takewhile (%a. a:int A)$schA)
-                         @@(Takewhile (%a. a:int B)$schB)
-                              @@(y\<leadsto>(mksch A B$xs
-                                       $(TL$(Dropwhile (%a. a:int A)$schA))
-                                       $(TL$(Dropwhile (%a. a:int B)$schB))
-                    )))
-              else
-                 ((Takewhile (%a. a:int A)$schA)
-                      @@ (y\<leadsto>(mksch A B$xs
-                              $(TL$(Dropwhile (%a. a:int A)$schA))
-                              $schB)))
-              )
-          else
-             (if y:act B then
-                 ((Takewhile (%a. a:int B)$schB)
-                       @@ (y\<leadsto>(mksch A B$xs
-                              $schA
-                              $(TL$(Dropwhile (%a. a:int B)$schB))
-                              )))
-             else
-               UU
-             )
-         )
-       ))"
-apply (rule trans)
-apply (rule fix_eq4)
-apply (rule mksch_def)
-apply (rule beta_cfun)
-apply simp
-done
+  "mksch A B =
+    (LAM tr schA schB.
+      case tr of
+        nil \<Rightarrow> nil
+      | x ## xs \<Rightarrow>
+          (case x of
+            UU \<Rightarrow> UU
+          | Def y \<Rightarrow>
+              (if y \<in> act A then
+                (if y \<in> act B then
+                  ((Takewhile (\<lambda>a. a \<in> int A) $ schA) @@
+                   (Takewhile (\<lambda>a. a \<in> int B) $ schB) @@
+                   (y \<leadsto> (mksch A B $ xs $(TL $ (Dropwhile (\<lambda>a. a \<in> int A) $ schA))
+                                         $(TL $ (Dropwhile (\<lambda>a. a \<in> int B) $ schB)))))
+                 else
+                  ((Takewhile (\<lambda>a. a \<in> int A) $ schA) @@
+                   (y \<leadsto> (mksch A B $ xs $ (TL $ (Dropwhile (\<lambda>a. a \<in> int A) $ schA)) $ schB))))
+               else
+                (if y \<in> act B then
+                  ((Takewhile (\<lambda>a. a \<in> int B) $ schB) @@
+                   (y \<leadsto> (mksch A B $ xs $ schA $ (TL $ (Dropwhile (\<lambda>a. a \<in> int B) $ schB)))))
+                 else UU))))"
+  apply (rule trans)
+  apply (rule fix_eq4)
+  apply (rule mksch_def)
+  apply (rule beta_cfun)
+  apply simp
+  done
+
+lemma mksch_UU: "mksch A B $ UU $ schA $ schB = UU"
+  apply (subst mksch_unfold)
+  apply simp
+  done
 
-lemma mksch_UU: "mksch A B$UU$schA$schB = UU"
-apply (subst mksch_unfold)
-apply simp
-done
-
-lemma mksch_nil: "mksch A B$nil$schA$schB = nil"
-apply (subst mksch_unfold)
-apply simp
-done
+lemma mksch_nil: "mksch A B $ nil $ schA $ schB = nil"
+  apply (subst mksch_unfold)
+  apply simp
+  done
 
-lemma mksch_cons1: "[|x:act A;x~:act B|]
-    ==> mksch A B$(x\<leadsto>tr)$schA$schB =
-          (Takewhile (%a. a:int A)$schA)
-          @@ (x\<leadsto>(mksch A B$tr$(TL$(Dropwhile (%a. a:int A)$schA))
-                              $schB))"
-apply (rule trans)
-apply (subst mksch_unfold)
-apply (simp add: Consq_def If_and_if)
-apply (simp add: Consq_def)
-done
+lemma mksch_cons1:
+  "x \<in> act A \<Longrightarrow> x \<notin> act B \<Longrightarrow>
+    mksch A B $ (x \<leadsto> tr) $ schA $ schB =
+      (Takewhile (\<lambda>a. a \<in> int A) $ schA) @@
+      (x \<leadsto> (mksch A B $ tr $ (TL $ (Dropwhile (\<lambda>a. a \<in> int A) $ schA)) $ schB))"
+  apply (rule trans)
+  apply (subst mksch_unfold)
+  apply (simp add: Consq_def If_and_if)
+  apply (simp add: Consq_def)
+  done
 
-lemma mksch_cons2: "[|x~:act A;x:act B|]
-    ==> mksch A B$(x\<leadsto>tr)$schA$schB =
-         (Takewhile (%a. a:int B)$schB)
-          @@ (x\<leadsto>(mksch A B$tr$schA$(TL$(Dropwhile (%a. a:int B)$schB))
-                             ))"
-apply (rule trans)
-apply (subst mksch_unfold)
-apply (simp add: Consq_def If_and_if)
-apply (simp add: Consq_def)
-done
+lemma mksch_cons2:
+  "x \<notin> act A \<Longrightarrow> x \<in> act B \<Longrightarrow>
+    mksch A B $ (x \<leadsto> tr) $ schA $ schB =
+      (Takewhile (\<lambda>a. a \<in> int B) $ schB) @@
+      (x \<leadsto> (mksch A B $ tr $ schA $ (TL $ (Dropwhile (\<lambda>a. a \<in> int B) $ schB))))"
+  apply (rule trans)
+  apply (subst mksch_unfold)
+  apply (simp add: Consq_def If_and_if)
+  apply (simp add: Consq_def)
+  done
 
-lemma mksch_cons3: "[|x:act A;x:act B|]
-    ==> mksch A B$(x\<leadsto>tr)$schA$schB =
-             (Takewhile (%a. a:int A)$schA)
-          @@ ((Takewhile (%a. a:int B)$schB)
-          @@ (x\<leadsto>(mksch A B$tr$(TL$(Dropwhile (%a. a:int A)$schA))
-                             $(TL$(Dropwhile (%a. a:int B)$schB))))
-              )"
-apply (rule trans)
-apply (subst mksch_unfold)
-apply (simp add: Consq_def If_and_if)
-apply (simp add: Consq_def)
-done
+lemma mksch_cons3:
+  "x \<in> act A \<Longrightarrow> x \<in> act B \<Longrightarrow>
+    mksch A B $ (x \<leadsto> tr) $ schA $ schB =
+      (Takewhile (\<lambda>a. a \<in> int A) $ schA) @@
+      ((Takewhile (\<lambda>a. a \<in> int B) $ schB) @@
+      (x \<leadsto> (mksch A B $ tr $ (TL $ (Dropwhile (\<lambda>a. a \<in> int A) $ schA))
+                            $ (TL $ (Dropwhile (\<lambda>a. a \<in> int B) $ schB)))))"
+  apply (rule trans)
+  apply (subst mksch_unfold)
+  apply (simp add: Consq_def If_and_if)
+  apply (simp add: Consq_def)
+  done
 
 lemmas compotr_simps = mksch_UU mksch_nil mksch_cons1 mksch_cons2 mksch_cons3
 
@@ -160,810 +139,802 @@
 
 subsection \<open>COMPOSITIONALITY on TRACE Level\<close>
 
-subsubsection "Lemmata for ==>"
+subsubsection "Lemmata for \<open>\<Longrightarrow>\<close>"
 
-(* Consequence out of ext1_ext2_is_not_act1(2), which in turn are consequences out of
-   the compatibility of IOA, in particular out of the condition that internals are
-   really hidden. *)
+text \<open>
+  Consequence out of \<open>ext1_ext2_is_not_act1(2)\<close>, which in turn are
+  consequences out of the compatibility of IOA, in particular out of the
+  condition that internals are really hidden.
+\<close>
 
-lemma compatibility_consequence1: "(eB & ~eA --> ~A) -->
-          (A & (eA | eB)) = (eA & A)"
-apply fast
-done
+lemma compatibility_consequence1: "(eB \<and> \<not> eA \<longrightarrow> \<not> A) \<longrightarrow> A \<and> (eA \<or> eB) \<longleftrightarrow> eA \<and> A"
+  by fast
 
 
 (* very similar to above, only the commutativity of | is used to make a slight change *)
-
-lemma compatibility_consequence2: "(eB & ~eA --> ~A) -->
-          (A & (eB | eA)) = (eA & A)"
-apply fast
-done
+lemma compatibility_consequence2: "(eB \<and> \<not> eA \<longrightarrow> \<not> A) \<longrightarrow> A \<and> (eB \<or> eA) \<longleftrightarrow> eA \<and> A"
+  by fast
 
 
-subsubsection "Lemmata for <=="
+subsubsection \<open>Lemmata for \<open>\<Longleftarrow>\<close>\<close>
 
 (* Lemma for substitution of looping assumption in another specific assumption *)
-lemma subst_lemma1: "[| f << (g x) ; x=(h x) |] ==> f << g (h x)"
-by (erule subst)
+lemma subst_lemma1: "f \<sqsubseteq> g x \<Longrightarrow> x = h x \<Longrightarrow> f \<sqsubseteq> g (h x)"
+  by (erule subst)
 
 (* Lemma for substitution of looping assumption in another specific assumption *)
-lemma subst_lemma2: "[| (f x) = y \<leadsto> g; x=(h x) |] ==> (f (h x)) = y \<leadsto> g"
-by (erule subst)
+lemma subst_lemma2: "(f x) = y \<leadsto> g \<Longrightarrow> x = h x \<Longrightarrow> f (h x) = y \<leadsto> g"
+  by (erule subst)
 
 lemma ForallAorB_mksch [rule_format]:
-  "!!A B. compatible A B ==>
-    ! schA schB. Forall (%x. x:act (A\<parallel>B)) tr
-    --> Forall (%x. x:act (A\<parallel>B)) (mksch A B$tr$schA$schB)"
-apply (tactic \<open>Seq_induct_tac @{context} "tr"
-  [@{thm Forall_def}, @{thm sforall_def}, @{thm mksch_def}] 1\<close>)
-apply auto
-apply (simp add: actions_of_par)
-apply (case_tac "a:act A")
-apply (case_tac "a:act B")
-(* a:A, a:B *)
-apply simp
-apply (rule Forall_Conc_impl [THEN mp])
-apply (simp add: intA_is_not_actB int_is_act)
-apply (rule Forall_Conc_impl [THEN mp])
-apply (simp add: intA_is_not_actB int_is_act)
-(* a:A,a~:B *)
-apply simp
-apply (rule Forall_Conc_impl [THEN mp])
-apply (simp add: intA_is_not_actB int_is_act)
-apply (case_tac "a:act B")
-(* a~:A, a:B *)
-apply simp
-apply (rule Forall_Conc_impl [THEN mp])
-apply (simp add: intA_is_not_actB int_is_act)
-(* a~:A,a~:B *)
-apply auto
-done
+  "compatible A B \<Longrightarrow>
+    \<forall>schA schB. Forall (\<lambda>x. x \<in> act (A \<parallel> B)) tr \<longrightarrow>
+      Forall (\<lambda>x. x \<in> act (A \<parallel> B)) (mksch A B $ tr $ schA $ schB)"
+  apply (tactic \<open>Seq_induct_tac @{context} "tr" @{thms Forall_def sforall_def mksch_def} 1\<close>)
+  apply auto
+  apply (simp add: actions_of_par)
+  apply (case_tac "a \<in> act A")
+  apply (case_tac "a \<in> act B")
+  text \<open>\<open>a \<in> A\<close>, \<open>a \<in> B\<close>\<close>
+  apply simp
+  apply (rule Forall_Conc_impl [THEN mp])
+  apply (simp add: intA_is_not_actB int_is_act)
+  apply (rule Forall_Conc_impl [THEN mp])
+  apply (simp add: intA_is_not_actB int_is_act)
+  text \<open>\<open>a \<in> A\<close>, \<open>a \<notin> B\<close>\<close>
+  apply simp
+  apply (rule Forall_Conc_impl [THEN mp])
+  apply (simp add: intA_is_not_actB int_is_act)
+  apply (case_tac "a:act B")
+  text \<open>\<open>a \<notin> A\<close>, \<open>a \<in> B\<close>\<close>
+  apply simp
+  apply (rule Forall_Conc_impl [THEN mp])
+  apply (simp add: intA_is_not_actB int_is_act)
+  text \<open>\<open>a \<notin> A\<close>, \<open>a \<notin> B\<close>\<close>
+  apply auto
+  done
 
-lemma ForallBnAmksch [rule_format (no_asm)]: "!!A B. compatible B A  ==>
-    ! schA schB.  (Forall (%x. x:act B & x~:act A) tr
-    --> Forall (%x. x:act B & x~:act A) (mksch A B$tr$schA$schB))"
-apply (tactic \<open>Seq_induct_tac @{context} "tr"
-  [@{thm Forall_def}, @{thm sforall_def}, @{thm mksch_def}] 1\<close>)
-apply auto
-apply (rule Forall_Conc_impl [THEN mp])
-apply (simp add: intA_is_not_actB int_is_act)
-done
+lemma ForallBnAmksch [rule_format]:
+  "compatible B A \<Longrightarrow>
+    \<forall>schA schB. Forall (\<lambda>x. x \<in> act B \<and> x \<notin> act A) tr \<longrightarrow>
+      Forall (\<lambda>x. x \<in> act B \<and> x \<notin> act A) (mksch A B $ tr $ schA $ schB)"
+  apply (tactic \<open>Seq_induct_tac @{context} "tr" @{thms Forall_def sforall_def mksch_def} 1\<close>)
+  apply auto
+  apply (rule Forall_Conc_impl [THEN mp])
+  apply (simp add: intA_is_not_actB int_is_act)
+  done
 
-lemma ForallAnBmksch [rule_format (no_asm)]: "!!A B. compatible A B ==>
-    ! schA schB.  (Forall (%x. x:act A & x~:act B) tr
-    --> Forall (%x. x:act A & x~:act B) (mksch A B$tr$schA$schB))"
-apply (tactic \<open>Seq_induct_tac @{context} "tr"
-  [@{thm Forall_def}, @{thm sforall_def}, @{thm mksch_def}] 1\<close>)
-apply auto
-apply (rule Forall_Conc_impl [THEN mp])
-apply (simp add: intA_is_not_actB int_is_act)
-done
+lemma ForallAnBmksch [rule_format]:
+  "compatible A B \<Longrightarrow>
+    \<forall>schA schB. Forall (\<lambda>x. x \<in> act A \<and> x \<notin> act B) tr \<longrightarrow>
+      Forall (\<lambda>x. x \<in> act A \<and> x \<notin> act B) (mksch A B $ tr $ schA $ schB)"
+  apply (tactic \<open>Seq_induct_tac @{context} "tr" @{thms Forall_def sforall_def mksch_def} 1\<close>)
+  apply auto
+  apply (rule Forall_Conc_impl [THEN mp])
+  apply (simp add: intA_is_not_actB int_is_act)
+  done
 
-(* safe-tac makes too many case distinctions with this lemma in the next proof *)
+(* safe_tac makes too many case distinctions with this lemma in the next proof *)
 declare FiniteConc [simp del]
 
-lemma FiniteL_mksch [rule_format (no_asm)]: "[| Finite tr; is_asig(asig_of A); is_asig(asig_of B) |] ==>
-    ! x y. Forall (%x. x:act A) x & Forall (%x. x:act B) y &
-           Filter (%a. a:ext A)$x = Filter (%a. a:act A)$tr &
-           Filter (%a. a:ext B)$y = Filter (%a. a:act B)$tr &
-           Forall (%x. x:ext (A\<parallel>B)) tr
-           --> Finite (mksch A B$tr$x$y)"
-
-apply (erule Seq_Finite_ind)
-apply simp
-(* main case *)
-apply simp
-apply auto
+lemma FiniteL_mksch [rule_format]:
+  "Finite tr \<Longrightarrow> is_asig (asig_of A) \<Longrightarrow> is_asig (asig_of B) \<Longrightarrow>
+    \<forall>x y.
+      Forall (\<lambda>x. x \<in> act A) x \<and> Forall (\<lambda>x. x \<in> act B) y \<and>
+      Filter (\<lambda>a. a \<in> ext A) $ x = Filter (\<lambda>a. a \<in> act A) $ tr \<and>
+      Filter (\<lambda>a. a \<in> ext B) $ y = Filter (\<lambda>a. a \<in> act B) $ tr \<and>
+      Forall (\<lambda>x. x \<in> ext (A \<parallel> B)) tr \<longrightarrow> Finite (mksch A B $ tr $ x $ y)"
+  apply (erule Seq_Finite_ind)
+  apply simp
+  text \<open>main case\<close>
+  apply simp
+  apply auto
 
-(* a: act A; a: act B *)
-apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
-apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
-back
-apply (erule conjE)+
-(* Finite (tw iA x) and Finite (tw iB y) *)
-apply (simp add: not_ext_is_int_or_not_act FiniteConc)
-(* now for conclusion IH applicable, but assumptions have to be transformed *)
-apply (drule_tac x = "x" and g = "Filter (%a. a:act A) $s" in subst_lemma2)
-apply assumption
-apply (drule_tac x = "y" and g = "Filter (%a. a:act B) $s" in subst_lemma2)
-apply assumption
-(* IH *)
-apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
+  text \<open>\<open>a \<in> act A\<close>; \<open>a \<in> act B\<close>\<close>
+  apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
+  apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
+  back
+  apply (erule conjE)+
+  text \<open>\<open>Finite (tw iA x)\<close> and \<open>Finite (tw iB y)\<close>\<close>
+  apply (simp add: not_ext_is_int_or_not_act FiniteConc)
+  text \<open>now for conclusion IH applicable, but assumptions have to be transformed\<close>
+  apply (drule_tac x = "x" and g = "Filter (\<lambda>a. a \<in> act A) $ s" in subst_lemma2)
+  apply assumption
+  apply (drule_tac x = "y" and g = "Filter (\<lambda>a. a \<in> act B) $ s" in subst_lemma2)
+  apply assumption
+  text \<open>IH\<close>
+  apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
 
-(* a: act B; a~: act A *)
-apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
+  text \<open>\<open>a \<in> act B\<close>; \<open>a \<notin> act A\<close>\<close>
+  apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
 
-apply (erule conjE)+
-(* Finite (tw iB y) *)
-apply (simp add: not_ext_is_int_or_not_act FiniteConc)
-(* now for conclusion IH applicable, but assumptions have to be transformed *)
-apply (drule_tac x = "y" and g = "Filter (%a. a:act B) $s" in subst_lemma2)
-apply assumption
-(* IH *)
-apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
+  apply (erule conjE)+
+  text \<open>\<open>Finite (tw iB y)\<close>\<close>
+  apply (simp add: not_ext_is_int_or_not_act FiniteConc)
+  text \<open>now for conclusion IH applicable, but assumptions have to be transformed\<close>
+  apply (drule_tac x = "y" and g = "Filter (\<lambda>a. a \<in> act B) $ s" in subst_lemma2)
+  apply assumption
+  text \<open>IH\<close>
+  apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
 
-(* a~: act B; a: act A *)
-apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
+  text \<open>\<open>a \<notin> act B\<close>; \<open>a \<in> act A\<close>\<close>
+  apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
 
-apply (erule conjE)+
-(* Finite (tw iA x) *)
-apply (simp add: not_ext_is_int_or_not_act FiniteConc)
-(* now for conclusion IH applicable, but assumptions have to be transformed *)
-apply (drule_tac x = "x" and g = "Filter (%a. a:act A) $s" in subst_lemma2)
-apply assumption
-(* IH *)
-apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
+  apply (erule conjE)+
+  text \<open>\<open>Finite (tw iA x)\<close>\<close>
+  apply (simp add: not_ext_is_int_or_not_act FiniteConc)
+  text \<open>now for conclusion IH applicable, but assumptions have to be transformed\<close>
+  apply (drule_tac x = "x" and g = "Filter (\<lambda>a. a \<in> act A) $ s" in subst_lemma2)
+  apply assumption
+  text \<open>IH\<close>
+  apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
 
-(* a~: act B; a~: act A *)
-apply (fastforce intro!: ext_is_act simp: externals_of_par)
-done
+  text \<open>\<open>a \<notin> act B\<close>; \<open>a \<notin> act A\<close>\<close>
+  apply (fastforce intro!: ext_is_act simp: externals_of_par)
+  done
 
 declare FiniteConc [simp]
 
 declare FilterConc [simp del]
 
-lemma reduceA_mksch1 [rule_format (no_asm)]: " [| Finite bs; is_asig(asig_of A); is_asig(asig_of B);compatible A B|] ==>
- ! y. Forall (%x. x:act B) y & Forall (%x. x:act B & x~:act A) bs &
-     Filter (%a. a:ext B)$y = Filter (%a. a:act B)$(bs @@ z)
-     --> (? y1 y2.  (mksch A B$(bs @@ z)$x$y) = (y1 @@ (mksch A B$z$x$y2)) &
-                    Forall (%x. x:act B & x~:act A) y1 &
-                    Finite y1 & y = (y1 @@ y2) &
-                    Filter (%a. a:ext B)$y1 = bs)"
-apply (frule_tac A1 = "A" in compat_commute [THEN iffD1])
-apply (erule Seq_Finite_ind)
-apply (rule allI)+
-apply (rule impI)
-apply (rule_tac x = "nil" in exI)
-apply (rule_tac x = "y" in exI)
-apply simp
-(* main case *)
-apply (rule allI)+
-apply (rule impI)
-apply simp
-apply (erule conjE)+
-apply simp
-(* divide_Seq on s *)
-apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
-apply (erule conjE)+
-(* transform assumption f eB y = f B (s@z) *)
-apply (drule_tac x = "y" and g = "Filter (%a. a:act B) $ (s@@z) " in subst_lemma2)
-apply assumption
-apply (simp add: not_ext_is_int_or_not_act FilterConc)
-(* apply IH *)
-apply (erule_tac x = "TL$ (Dropwhile (%a. a:int B) $y) " in allE)
-apply (simp add: ForallTL ForallDropwhile FilterConc)
-apply (erule exE)+
-apply (erule conjE)+
-apply (simp add: FilterConc)
-(* for replacing IH in conclusion *)
-apply (rotate_tac -2)
-(* instantiate y1a and y2a *)
-apply (rule_tac x = "Takewhile (%a. a:int B) $y @@ a\<leadsto>y1" in exI)
-apply (rule_tac x = "y2" in exI)
-(* elminate all obligations up to two depending on Conc_assoc *)
-apply (simp add: intA_is_not_actB int_is_act int_is_not_ext FilterConc)
-apply (simp (no_asm) add: Conc_assoc FilterConc)
-done
+lemma reduceA_mksch1 [rule_format]:
+  "Finite bs \<Longrightarrow> is_asig (asig_of A) \<Longrightarrow> is_asig (asig_of B) \<Longrightarrow> compatible A B \<Longrightarrow>
+    \<forall>y. Forall (\<lambda>x. x \<in> act B) y \<and> Forall (\<lambda>x. x \<in> act B \<and> x \<notin> act A) bs \<and>
+      Filter (\<lambda>a. a \<in> ext B) $ y = Filter (\<lambda>a. a \<in> act B) $ (bs @@ z) \<longrightarrow>
+      (\<exists>y1 y2.
+        (mksch A B $ (bs @@ z) $ x $ y) = (y1 @@ (mksch A B $ z $ x $ y2)) \<and>
+        Forall (\<lambda>x. x \<in> act B \<and> x \<notin> act A) y1 \<and>
+        Finite y1 \<and> y = (y1 @@ y2) \<and>
+        Filter (\<lambda>a. a \<in> ext B) $ y1 = bs)"
+  apply (frule_tac A1 = "A" in compat_commute [THEN iffD1])
+  apply (erule Seq_Finite_ind)
+  apply (rule allI)+
+  apply (rule impI)
+  apply (rule_tac x = "nil" in exI)
+  apply (rule_tac x = "y" in exI)
+  apply simp
+  text \<open>main case\<close>
+  apply (rule allI)+
+  apply (rule impI)
+  apply simp
+  apply (erule conjE)+
+  apply simp
+  text \<open>\<open>divide_Seq\<close> on \<open>s\<close>\<close>
+  apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
+  apply (erule conjE)+
+  text \<open>transform assumption \<open>f eB y = f B (s @ z)\<close>\<close>
+  apply (drule_tac x = "y" and g = "Filter (\<lambda>a. a \<in> act B) $ (s @@ z) " in subst_lemma2)
+  apply assumption
+  apply (simp add: not_ext_is_int_or_not_act FilterConc)
+  text \<open>apply IH\<close>
+  apply (erule_tac x = "TL $ (Dropwhile (\<lambda>a. a \<in> int B) $ y)" in allE)
+  apply (simp add: ForallTL ForallDropwhile FilterConc)
+  apply (erule exE)+
+  apply (erule conjE)+
+  apply (simp add: FilterConc)
+  text \<open>for replacing IH in conclusion\<close>
+  apply (rotate_tac -2)
+  text \<open>instantiate \<open>y1a\<close> and \<open>y2a\<close>\<close>
+  apply (rule_tac x = "Takewhile (\<lambda>a. a \<in> int B) $ y @@ a \<leadsto> y1" in exI)
+  apply (rule_tac x = "y2" in exI)
+  text \<open>elminate all obligations up to two depending on \<open>Conc_assoc\<close>\<close>
+  apply (simp add: intA_is_not_actB int_is_act int_is_not_ext FilterConc)
+  apply (simp add: Conc_assoc FilterConc)
+  done
 
 lemmas reduceA_mksch = conjI [THEN [6] conjI [THEN [5] reduceA_mksch1]]
 
 lemma reduceB_mksch1 [rule_format]:
-" [| Finite a_s; is_asig(asig_of A); is_asig(asig_of B);compatible A B|] ==>
- ! x. Forall (%x. x:act A) x & Forall (%x. x:act A & x~:act B) a_s &
-     Filter (%a. a:ext A)$x = Filter (%a. a:act A)$(a_s @@ z)
-     --> (? x1 x2.  (mksch A B$(a_s @@ z)$x$y) = (x1 @@ (mksch A B$z$x2$y)) &
-                    Forall (%x. x:act A & x~:act B) x1 &
-                    Finite x1 & x = (x1 @@ x2) &
-                    Filter (%a. a:ext A)$x1 = a_s)"
-apply (frule_tac A1 = "A" in compat_commute [THEN iffD1])
-apply (erule Seq_Finite_ind)
-apply (rule allI)+
-apply (rule impI)
-apply (rule_tac x = "nil" in exI)
-apply (rule_tac x = "x" in exI)
-apply simp
-(* main case *)
-apply (rule allI)+
-apply (rule impI)
-apply simp
-apply (erule conjE)+
-apply simp
-(* divide_Seq on s *)
-apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
-apply (erule conjE)+
-(* transform assumption f eA x = f A (s@z) *)
-apply (drule_tac x = "x" and g = "Filter (%a. a:act A) $ (s@@z) " in subst_lemma2)
-apply assumption
-apply (simp add: not_ext_is_int_or_not_act FilterConc)
-(* apply IH *)
-apply (erule_tac x = "TL$ (Dropwhile (%a. a:int A) $x) " in allE)
-apply (simp add: ForallTL ForallDropwhile FilterConc)
-apply (erule exE)+
-apply (erule conjE)+
-apply (simp add: FilterConc)
-(* for replacing IH in conclusion *)
-apply (rotate_tac -2)
-(* instantiate y1a and y2a *)
-apply (rule_tac x = "Takewhile (%a. a:int A) $x @@ a\<leadsto>x1" in exI)
-apply (rule_tac x = "x2" in exI)
-(* elminate all obligations up to two depending on Conc_assoc *)
-apply (simp add: intA_is_not_actB int_is_act int_is_not_ext FilterConc)
-apply (simp (no_asm) add: Conc_assoc FilterConc)
-done
+  "Finite a_s \<Longrightarrow> is_asig (asig_of A) \<Longrightarrow> is_asig (asig_of B) \<Longrightarrow> compatible A B \<Longrightarrow>
+    \<forall>x. Forall (\<lambda>x. x \<in> act A) x \<and> Forall (\<lambda>x. x \<in> act A \<and> x \<notin> act B) a_s \<and>
+      Filter (\<lambda>a. a \<in> ext A) $ x = Filter (\<lambda>a. a \<in> act A) $ (a_s @@ z) \<longrightarrow>
+      (\<exists>x1 x2.
+        (mksch A B $ (a_s @@ z) $ x $ y) = (x1 @@ (mksch A B $ z $ x2 $ y)) \<and>
+        Forall (\<lambda>x. x \<in> act A \<and> x \<notin> act B) x1 \<and>
+        Finite x1 \<and> x = (x1 @@ x2) \<and>
+        Filter (\<lambda>a. a \<in> ext A) $ x1 = a_s)"
+  apply (frule_tac A1 = "A" in compat_commute [THEN iffD1])
+  apply (erule Seq_Finite_ind)
+  apply (rule allI)+
+  apply (rule impI)
+  apply (rule_tac x = "nil" in exI)
+  apply (rule_tac x = "x" in exI)
+  apply simp
+  text \<open>main case\<close>
+  apply (rule allI)+
+  apply (rule impI)
+  apply simp
+  apply (erule conjE)+
+  apply simp
+  text \<open>\<open>divide_Seq\<close> on \<open>s\<close>\<close>
+  apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
+  apply (erule conjE)+
+  text \<open>transform assumption \<open>f eA x = f A (s @ z)\<close>\<close>
+  apply (drule_tac x = "x" and g = "Filter (\<lambda>a. a \<in> act A) $ (s @@ z)" in subst_lemma2)
+  apply assumption
+  apply (simp add: not_ext_is_int_or_not_act FilterConc)
+  text \<open>apply IH\<close>
+  apply (erule_tac x = "TL $ (Dropwhile (\<lambda>a. a \<in> int A) $ x)" in allE)
+  apply (simp add: ForallTL ForallDropwhile FilterConc)
+  apply (erule exE)+
+  apply (erule conjE)+
+  apply (simp add: FilterConc)
+  text \<open>for replacing IH in conclusion\<close>
+  apply (rotate_tac -2)
+  text \<open>instantiate \<open>y1a\<close> and \<open>y2a\<close>\<close>
+  apply (rule_tac x = "Takewhile (\<lambda>a. a \<in> int A) $ x @@ a \<leadsto> x1" in exI)
+  apply (rule_tac x = "x2" in exI)
+  text \<open>eliminate all obligations up to two depending on \<open>Conc_assoc\<close>\<close>
+  apply (simp add: intA_is_not_actB int_is_act int_is_not_ext FilterConc)
+  apply (simp add: Conc_assoc FilterConc)
+  done
 
 lemmas reduceB_mksch = conjI [THEN [6] conjI [THEN [5] reduceB_mksch1]]
 
 declare FilterConc [simp]
 
 
-subsection "Filtering external actions out of mksch(tr,schA,schB) yields the oracle tr"
+subsection \<open>Filtering external actions out of \<open>mksch (tr, schA, schB)\<close> yields the oracle \<open>tr\<close>\<close>
 
 lemma FilterA_mksch_is_tr:
-"!! A B. [| compatible A B; compatible B A;
-            is_asig(asig_of A); is_asig(asig_of B) |] ==>
-  ! schA schB. Forall (%x. x:act A) schA & Forall (%x. x:act B) schB &
-  Forall (%x. x:ext (A\<parallel>B)) tr &
-  Filter (%a. a:act A)$tr << Filter (%a. a:ext A)$schA &
-  Filter (%a. a:act B)$tr << Filter (%a. a:ext B)$schB
-  --> Filter (%a. a:ext (A\<parallel>B))$(mksch A B$tr$schA$schB) = tr"
-
-apply (tactic \<open>Seq_induct_tac @{context} "tr"
-  [@{thm Forall_def}, @{thm sforall_def}, @{thm mksch_def}] 1\<close>)
-(* main case *)
-(* splitting into 4 cases according to a:A, a:B *)
-apply auto
+  "compatible A B \<Longrightarrow> compatible B A \<Longrightarrow> is_asig (asig_of A) \<Longrightarrow> is_asig (asig_of B) \<Longrightarrow>
+    \<forall>schA schB.
+      Forall (\<lambda>x. x \<in> act A) schA \<and>
+      Forall (\<lambda>x. x \<in> act B) schB \<and>
+      Forall (\<lambda>x. x \<in> ext (A \<parallel> B)) tr \<and>
+      Filter (\<lambda>a. a \<in> act A) $ tr \<sqsubseteq> Filter (\<lambda>a. a \<in> ext A) $ schA \<and>
+      Filter (\<lambda>a. a \<in> act B) $ tr \<sqsubseteq> Filter (\<lambda>a. a \<in> ext B) $ schB
+      \<longrightarrow> Filter (\<lambda>a. a \<in> ext (A \<parallel> B)) $ (mksch A B $ tr $ schA $ schB) = tr"
+  apply (tactic \<open>Seq_induct_tac @{context} "tr" @{thms Forall_def sforall_def mksch_def} 1\<close>)
+  text \<open>main case\<close>
+  text \<open>splitting into 4 cases according to \<open>a \<in> A\<close>, \<open>a \<in> B\<close>\<close>
+  apply auto
 
-(* Case a:A, a:B *)
-apply (frule divide_Seq)
-apply (frule divide_Seq)
-back
-apply (erule conjE)+
-(* filtering internals of A in schA and of B in schB is nil *)
-apply (simp add: not_ext_is_int_or_not_act externals_of_par intA_is_not_extB int_is_not_ext)
-(* conclusion of IH ok, but assumptions of IH have to be transformed *)
-apply (drule_tac x = "schA" in subst_lemma1)
-apply assumption
-apply (drule_tac x = "schB" in subst_lemma1)
-apply assumption
-(* IH *)
-apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
+  text \<open>Case \<open>a \<in> A\<close>, \<open>a \<in> B\<close>\<close>
+  apply (frule divide_Seq)
+  apply (frule divide_Seq)
+  back
+  apply (erule conjE)+
+  text \<open>filtering internals of \<open>A\<close> in \<open>schA\<close> and of \<open>B\<close> in \<open>schB\<close> is \<open>nil\<close>\<close>
+  apply (simp add: not_ext_is_int_or_not_act externals_of_par intA_is_not_extB int_is_not_ext)
+  text \<open>conclusion of IH ok, but assumptions of IH have to be transformed\<close>
+  apply (drule_tac x = "schA" in subst_lemma1)
+  apply assumption
+  apply (drule_tac x = "schB" in subst_lemma1)
+  apply assumption
+  text \<open>IH\<close>
+  apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
 
-(* Case a:A, a~:B *)
-apply (frule divide_Seq)
-apply (erule conjE)+
-(* filtering internals of A is nil *)
-apply (simp add: not_ext_is_int_or_not_act externals_of_par intA_is_not_extB int_is_not_ext)
-apply (drule_tac x = "schA" in subst_lemma1)
-apply assumption
-apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
+  text \<open>Case \<open>a \<in> A\<close>, \<open>a \<notin> B\<close>\<close>
+  apply (frule divide_Seq)
+  apply (erule conjE)+
+  text \<open>filtering internals of \<open>A\<close> is \<open>nil\<close>\<close>
+  apply (simp add: not_ext_is_int_or_not_act externals_of_par intA_is_not_extB int_is_not_ext)
+  apply (drule_tac x = "schA" in subst_lemma1)
+  apply assumption
+  apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
 
-(* Case a:B, a~:A *)
-apply (frule divide_Seq)
-apply (erule conjE)+
-(* filtering internals of A is nil *)
-apply (simp add: not_ext_is_int_or_not_act externals_of_par intA_is_not_extB int_is_not_ext)
-apply (drule_tac x = "schB" in subst_lemma1)
-back
-apply assumption
-apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
+  text \<open>Case \<open>a \<in> B\<close>, \<open>a \<notin> A\<close>\<close>
+  apply (frule divide_Seq)
+  apply (erule conjE)+
+  text \<open>filtering internals of \<open>A\<close> is \<open>nil\<close>\<close>
+  apply (simp add: not_ext_is_int_or_not_act externals_of_par intA_is_not_extB int_is_not_ext)
+  apply (drule_tac x = "schB" in subst_lemma1)
+  back
+  apply assumption
+  apply (simp add: not_ext_is_int_or_not_act ForallTL ForallDropwhile)
 
-(* Case a~:A, a~:B *)
-apply (fastforce intro!: ext_is_act simp: externals_of_par)
-done
+  text \<open>Case \<open>a \<notin> A\<close>, \<open>a \<notin> B\<close>\<close>
+  apply (fastforce intro!: ext_is_act simp: externals_of_par)
+  done
 
 
 subsection" Filter of mksch(tr,schA,schB) to A is schA -- take lemma proof"
 
-lemma FilterAmksch_is_schA: "!! A B. [| compatible A B; compatible B A;
-  is_asig(asig_of A); is_asig(asig_of B) |] ==>
-  Forall (%x. x:ext (A\<parallel>B)) tr &
-  Forall (%x. x:act A) schA & Forall (%x. x:act B) schB &
-  Filter (%a. a:ext A)$schA = Filter (%a. a:act A)$tr &
-  Filter (%a. a:ext B)$schB = Filter (%a. a:act B)$tr &
-  LastActExtsch A schA & LastActExtsch B schB
-  --> Filter (%a. a:act A)$(mksch A B$tr$schA$schB) = schA"
-apply (intro strip)
-apply (rule seq.take_lemma)
-apply (rule mp)
-prefer 2 apply assumption
-back back back back
-apply (rule_tac x = "schA" in spec)
-apply (rule_tac x = "schB" in spec)
-apply (rule_tac x = "tr" in spec)
-apply (tactic "thin_tac' @{context} 5 1")
-apply (rule nat_less_induct)
-apply (rule allI)+
-apply (rename_tac tr schB schA)
-apply (intro strip)
-apply (erule conjE)+
+lemma FilterAmksch_is_schA:
+  "compatible A B \<Longrightarrow> compatible B A \<Longrightarrow> is_asig (asig_of A) \<Longrightarrow> is_asig (asig_of B) \<Longrightarrow>
+    Forall (\<lambda>x. x \<in> ext (A \<parallel> B)) tr \<and>
+    Forall (\<lambda>x. x \<in> act A) schA \<and>
+    Forall (\<lambda>x. x \<in> act B) schB \<and>
+    Filter (\<lambda>a. a \<in> ext A) $ schA = Filter (\<lambda>a. a \<in> act A) $ tr \<and>
+    Filter (\<lambda>a. a \<in> ext B) $ schB = Filter (\<lambda>a. a \<in> act B) $ tr \<and>
+    LastActExtsch A schA \<and> LastActExtsch B schB
+    \<longrightarrow> Filter (\<lambda>a. a \<in> act A) $ (mksch A B $ tr $ schA $ schB) = schA"
+  apply (intro strip)
+  apply (rule seq.take_lemma)
+  apply (rule mp)
+  prefer 2 apply assumption
+  back back back back
+  apply (rule_tac x = "schA" in spec)
+  apply (rule_tac x = "schB" in spec)
+  apply (rule_tac x = "tr" in spec)
+  apply (tactic "thin_tac' @{context} 5 1")
+  apply (rule nat_less_induct)
+  apply (rule allI)+
+  apply (rename_tac tr schB schA)
+  apply (intro strip)
+  apply (erule conjE)+
 
-apply (case_tac "Forall (%x. x:act B & x~:act A) tr")
+  apply (case_tac "Forall (\<lambda>x. x \<in> act B \<and> x \<notin> act A) tr")
 
-apply (rule seq_take_lemma [THEN iffD2, THEN spec])
-apply (tactic "thin_tac' @{context} 5 1")
+  apply (rule seq_take_lemma [THEN iffD2, THEN spec])
+  apply (tactic "thin_tac' @{context} 5 1")
 
 
-apply (case_tac "Finite tr")
+  apply (case_tac "Finite tr")
 
-(* both sides of this equation are nil *)
-apply (subgoal_tac "schA=nil")
-apply (simp (no_asm_simp))
-(* first side: mksch = nil *)
-apply (auto intro!: ForallQFilterPnil ForallBnAmksch FiniteL_mksch)[1]
-(* second side: schA = nil *)
-apply (erule_tac A = "A" in LastActExtimplnil)
-apply (simp (no_asm_simp))
-apply (erule_tac Q = "%x. x:act B & x~:act A" in ForallQFilterPnil)
-apply assumption
-apply fast
+  text \<open>both sides of this equation are \<open>nil\<close>\<close>
+  apply (subgoal_tac "schA = nil")
+  apply simp
+  text \<open>first side: \<open>mksch = nil\<close>\<close>
+  apply (auto intro!: ForallQFilterPnil ForallBnAmksch FiniteL_mksch)[1]
+  text \<open>second side: \<open>schA = nil\<close>\<close>
+  apply (erule_tac A = "A" in LastActExtimplnil)
+  apply simp
+  apply (erule_tac Q = "\<lambda>x. x \<in> act B \<and> x \<notin> act A" in ForallQFilterPnil)
+  apply assumption
+  apply fast
 
-(* case ~ Finite s *)
+  text \<open>case \<open>\<not> Finite s\<close>\<close>
 
-(* both sides of this equation are UU *)
-apply (subgoal_tac "schA=UU")
-apply (simp (no_asm_simp))
-(* first side: mksch = UU *)
-apply (auto intro!: ForallQFilterPUU finiteR_mksch' ForallBnAmksch)[1]
-(* schA = UU *)
-apply (erule_tac A = "A" in LastActExtimplUU)
-apply (simp (no_asm_simp))
-apply (erule_tac Q = "%x. x:act B & x~:act A" in ForallQFilterPUU)
-apply assumption
-apply fast
+  text \<open>both sides of this equation are \<open>UU\<close>\<close>
+  apply (subgoal_tac "schA = UU")
+  apply simp
+  text \<open>first side: \<open>mksch = UU\<close>\<close>
+  apply (auto intro!: ForallQFilterPUU finiteR_mksch' ForallBnAmksch)[1]
+  text \<open>\<open>schA = UU\<close>\<close>
+  apply (erule_tac A = "A" in LastActExtimplUU)
+  apply simp
+  apply (erule_tac Q = "\<lambda>x. x \<in> act B \<and> x \<notin> act A" in ForallQFilterPUU)
+  apply assumption
+  apply fast
 
-(* case" ~ Forall (%x.x:act B & x~:act A) s" *)
+  text \<open>case \<open>\<not> Forall (\<lambda>x. x \<in> act B \<and> x \<notin> act A) s\<close>\<close>
 
-apply (drule divide_Seq3)
+  apply (drule divide_Seq3)
 
-apply (erule exE)+
-apply (erule conjE)+
-apply hypsubst
+  apply (erule exE)+
+  apply (erule conjE)+
+  apply hypsubst
 
-(* bring in lemma reduceA_mksch *)
-apply (frule_tac x = "schA" and y = "schB" and A = "A" and B = "B" in reduceA_mksch)
-apply assumption+
-apply (erule exE)+
-apply (erule conjE)+
+  text \<open>bring in lemma \<open>reduceA_mksch\<close>\<close>
+  apply (frule_tac x = "schA" and y = "schB" and A = "A" and B = "B" in reduceA_mksch)
+  apply assumption+
+  apply (erule exE)+
+  apply (erule conjE)+
 
-(* use reduceA_mksch to rewrite conclusion *)
-apply hypsubst
-apply simp
+  text \<open>use \<open>reduceA_mksch\<close> to rewrite conclusion\<close>
+  apply hypsubst
+  apply simp
 
-(* eliminate the B-only prefix *)
+  text \<open>eliminate the \<open>B\<close>-only prefix\<close>
 
-apply (subgoal_tac " (Filter (%a. a :act A) $y1) = nil")
-apply (erule_tac [2] ForallQFilterPnil)
-prefer 2 apply assumption
-prefer 2 apply fast
+  apply (subgoal_tac "(Filter (\<lambda>a. a \<in> act A) $ y1) = nil")
+  apply (erule_tac [2] ForallQFilterPnil)
+  prefer 2 apply assumption
+  prefer 2 apply fast
 
-(* Now real recursive step follows (in y) *)
+  text \<open>Now real recursive step follows (in \<open>y\<close>)\<close>
 
-apply simp
-apply (case_tac "x:act A")
-apply (case_tac "x~:act B")
-apply (rotate_tac -2)
-apply simp
+  apply simp
+  apply (case_tac "x \<in> act A")
+  apply (case_tac "x \<notin> act B")
+  apply (rotate_tac -2)
+  apply simp
 
-apply (subgoal_tac "Filter (%a. a:act A & a:ext B) $y1=nil")
-apply (rotate_tac -1)
-apply simp
-(* eliminate introduced subgoal 2 *)
-apply (erule_tac [2] ForallQFilterPnil)
-prefer 2 apply assumption
-prefer 2 apply fast
+  apply (subgoal_tac "Filter (\<lambda>a. a \<in> act A \<and> a \<in> ext B) $ y1 = nil")
+  apply (rotate_tac -1)
+  apply simp
+  text \<open>eliminate introduced subgoal 2\<close>
+  apply (erule_tac [2] ForallQFilterPnil)
+  prefer 2 apply assumption
+  prefer 2 apply fast
 
-(* bring in divide Seq for s *)
-apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
-apply (erule conjE)+
+  text \<open>bring in \<open>divide_Seq\<close> for \<open>s\<close>\<close>
+  apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
+  apply (erule conjE)+
 
-(* subst divide_Seq in conclusion, but only at the righest occurrence *)
-apply (rule_tac t = "schA" in ssubst)
-back
-back
-back
-apply assumption
+  text \<open>subst \<open>divide_Seq\<close> in conclusion, but only at the rightest occurrence\<close>
+  apply (rule_tac t = "schA" in ssubst)
+  back
+  back
+  back
+  apply assumption
 
-(* reduce trace_takes from n to strictly smaller k *)
-apply (rule take_reduction)
+  text \<open>reduce trace_takes from \<open>n\<close> to strictly smaller \<open>k\<close>\<close>
+  apply (rule take_reduction)
 
-(* f A (tw iA) = tw ~eA *)
-apply (simp add: int_is_act not_ext_is_int_or_not_act)
-apply (rule refl)
-apply (simp add: int_is_act not_ext_is_int_or_not_act)
-apply (rotate_tac -11)
+  text \<open>\<open>f A (tw iA) = tw \<not> eA\<close>\<close>
+  apply (simp add: int_is_act not_ext_is_int_or_not_act)
+  apply (rule refl)
+  apply (simp add: int_is_act not_ext_is_int_or_not_act)
+  apply (rotate_tac -11)
 
-(* now conclusion fulfills induction hypothesis, but assumptions are not ready *)
+  text \<open>now conclusion fulfills induction hypothesis, but assumptions are not ready\<close>
 
-(* assumption Forall tr *)
-(* assumption schB *)
-apply (simp add: ext_and_act)
-(* assumption schA *)
-apply (drule_tac x = "schA" and g = "Filter (%a. a:act A) $rs" in subst_lemma2)
-apply assumption
-apply (simp add: int_is_not_ext)
-(* assumptions concerning LastActExtsch, cannot be rewritten, as LastActExtsmalli are looping  *)
-apply (drule_tac sch = "schA" and P = "%a. a:int A" in LastActExtsmall1)
-apply (frule_tac ?sch1.0 = "y1" in LastActExtsmall2)
-apply assumption
+  text \<open>assumption \<open>Forall tr\<close>\<close>
+  text \<open>assumption \<open>schB\<close>\<close>
+  apply (simp add: ext_and_act)
+  text \<open>assumption \<open>schA\<close>\<close>
+  apply (drule_tac x = "schA" and g = "Filter (\<lambda>a. a \<in> act A) $ rs" in subst_lemma2)
+  apply assumption
+  apply (simp add: int_is_not_ext)
+  text \<open>assumptions concerning \<open>LastActExtsch\<close>, cannot be rewritten, as \<open>LastActExtsmalli\<close> are looping\<close>
+  apply (drule_tac sch = "schA" and P = "\<lambda>a. a \<in> int A" in LastActExtsmall1)
+  apply (frule_tac ?sch1.0 = "y1" in LastActExtsmall2)
+  apply assumption
 
-(* assumption Forall schA *)
-apply (drule_tac s = "schA" and P = "Forall (%x. x:act A) " in subst)
-apply assumption
-apply (simp add: int_is_act)
+  text \<open>assumption \<open>Forall schA\<close>\<close>
+  apply (drule_tac s = "schA" and P = "Forall (\<lambda>x. x \<in> act A) " in subst)
+  apply assumption
+  apply (simp add: int_is_act)
 
-(* case x:actions(asig_of A) & x: actions(asig_of B) *)
-
+  text \<open>case \<open>x \<in> actions (asig_of A) \<and> x \<in> actions (asig_of B)\<close>\<close>
 
-apply (rotate_tac -2)
-apply simp
+  apply (rotate_tac -2)
+  apply simp
 
-apply (subgoal_tac "Filter (%a. a:act A & a:ext B) $y1=nil")
-apply (rotate_tac -1)
-apply simp
-(* eliminate introduced subgoal 2 *)
-apply (erule_tac [2] ForallQFilterPnil)
-prefer 2 apply (assumption)
-prefer 2 apply (fast)
+  apply (subgoal_tac "Filter (\<lambda>a. a \<in> act A \<and> a \<in> ext B) $ y1 = nil")
+  apply (rotate_tac -1)
+  apply simp
+  text \<open>eliminate introduced subgoal 2\<close>
+  apply (erule_tac [2] ForallQFilterPnil)
+  prefer 2 apply assumption
+  prefer 2 apply fast
 
-(* bring in divide Seq for s *)
-apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
-apply (erule conjE)+
+  text \<open>bring in \<open>divide_Seq\<close> for \<open>s\<close>\<close>
+  apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
+  apply (erule conjE)+
 
-(* subst divide_Seq in conclusion, but only at the rightmost occurrence *)
-apply (rule_tac t = "schA" in ssubst)
-back
-back
-back
-apply assumption
+  text \<open>subst \<open>divide_Seq\<close> in conclusion, but only at the rightmost occurrence\<close>
+  apply (rule_tac t = "schA" in ssubst)
+  back
+  back
+  back
+  apply assumption
 
-(* f A (tw iA) = tw ~eA *)
-apply (simp add: int_is_act not_ext_is_int_or_not_act)
+  text \<open>\<open>f A (tw iA) = tw \<not> eA\<close>\<close>
+  apply (simp add: int_is_act not_ext_is_int_or_not_act)
 
-(* rewrite assumption forall and schB *)
-apply (rotate_tac 13)
-apply (simp add: ext_and_act)
+  text \<open>rewrite assumption forall and \<open>schB\<close>\<close>
+  apply (rotate_tac 13)
+  apply (simp add: ext_and_act)
 
-(* divide_Seq for schB2 *)
-apply (frule_tac y = "y2" in sym [THEN eq_imp_below, THEN divide_Seq])
-apply (erule conjE)+
-(* assumption schA *)
-apply (drule_tac x = "schA" and g = "Filter (%a. a:act A) $rs" in subst_lemma2)
-apply assumption
-apply (simp add: int_is_not_ext)
+  text \<open>\<open>divide_Seq\<close> for \<open>schB2\<close>\<close>
+  apply (frule_tac y = "y2" in sym [THEN eq_imp_below, THEN divide_Seq])
+  apply (erule conjE)+
+  text \<open>assumption \<open>schA\<close>\<close>
+  apply (drule_tac x = "schA" and g = "Filter (%a. a:act A) $rs" in subst_lemma2)
+  apply assumption
+  apply (simp add: int_is_not_ext)
 
-(* f A (tw iB schB2) = nil *)
-apply (simp add: int_is_not_ext not_ext_is_int_or_not_act intA_is_not_actB)
+  text \<open>\<open>f A (tw iB schB2) = nil\<close>\<close>
+  apply (simp add: int_is_not_ext not_ext_is_int_or_not_act intA_is_not_actB)
 
 
-(* reduce trace_takes from n to strictly smaller k *)
-apply (rule take_reduction)
-apply (rule refl)
-apply (rule refl)
-
-(* now conclusion fulfills induction hypothesis, but assumptions are not all ready *)
+  text \<open>reduce trace_takes from \<open>n\<close> to strictly smaller \<open>k\<close>\<close>
+  apply (rule take_reduction)
+  apply (rule refl)
+  apply (rule refl)
 
-(* assumption schB *)
-apply (drule_tac x = "y2" and g = "Filter (%a. a:act B) $rs" in subst_lemma2)
-apply assumption
-apply (simp add: intA_is_not_actB int_is_not_ext)
+  text \<open>now conclusion fulfills induction hypothesis, but assumptions are not all ready\<close>
 
-(* conclusions concerning LastActExtsch, cannot be rewritten, as LastActExtsmalli are looping  *)
-apply (drule_tac sch = "schA" and P = "%a. a:int A" in LastActExtsmall1)
-apply (frule_tac ?sch1.0 = "y1" in LastActExtsmall2)
-apply assumption
-apply (drule_tac sch = "y2" and P = "%a. a:int B" in LastActExtsmall1)
+  text \<open>assumption \<open>schB\<close>\<close>
+  apply (drule_tac x = "y2" and g = "Filter (\<lambda>a. a \<in> act B) $ rs" in subst_lemma2)
+  apply assumption
+  apply (simp add: intA_is_not_actB int_is_not_ext)
 
-(* assumption Forall schA, and Forall schA are performed by ForallTL,ForallDropwhile *)
-apply (simp add: ForallTL ForallDropwhile)
+  text \<open>conclusions concerning \<open>LastActExtsch\<close>, cannot be rewritten, as \<open>LastActExtsmalli\<close> are looping\<close>
+  apply (drule_tac sch = "schA" and P = "\<lambda>a. a \<in> int A" in LastActExtsmall1)
+  apply (frule_tac ?sch1.0 = "y1" in LastActExtsmall2)
+  apply assumption
+  apply (drule_tac sch = "y2" and P = "\<lambda>a. a \<in> int B" in LastActExtsmall1)
 
-(* case x~:A & x:B  *)
-(* cannot occur, as just this case has been scheduled out before as the B-only prefix *)
-apply (case_tac "x:act B")
-apply fast
+  text \<open>assumption \<open>Forall schA\<close>, and \<open>Forall schA\<close> are performed by \<open>ForallTL\<close>, \<open>ForallDropwhile\<close>\<close>
+  apply (simp add: ForallTL ForallDropwhile)
 
-(* case x~:A & x~:B  *)
-(* cannot occur because of assumption: Forall (a:ext A | a:ext B) *)
-apply (rotate_tac -9)
-(* reduce forall assumption from tr to (x\<leadsto>rs) *)
-apply (simp add: externals_of_par)
-apply (fast intro!: ext_is_act)
+  text \<open>case \<open>x \<notin> A \<and> x \<in> B\<close>\<close>
+  text \<open>cannot occur, as just this case has been scheduled out before as the \<open>B\<close>-only prefix\<close>
+  apply (case_tac "x \<in> act B")
+  apply fast
 
-done
-
+  text \<open>case \<open>x \<notin> A \<and> x \<notin> B\<close>\<close>
+  text \<open>cannot occur because of assumption: \<open>Forall (a \<in> ext A \<or> a \<in> ext B)\<close>\<close>
+  apply (rotate_tac -9)
+  text \<open>reduce forall assumption from \<open>tr\<close> to \<open>x \<leadsto> rs\<close>\<close>
+  apply (simp add: externals_of_par)
+  apply (fast intro!: ext_is_act)
+  done
 
 
-subsection" Filter of mksch(tr,schA,schB) to B is schB -- take lemma proof"
+subsection \<open>Filter of \<open>mksch (tr, schA, schB)\<close> to \<open>B\<close> is \<open>schB\<close> -- take lemma proof\<close>
 
-lemma FilterBmksch_is_schB: "!! A B. [| compatible A B; compatible B A;
-  is_asig(asig_of A); is_asig(asig_of B) |] ==>
-  Forall (%x. x:ext (A\<parallel>B)) tr &
-  Forall (%x. x:act A) schA & Forall (%x. x:act B) schB &
-  Filter (%a. a:ext A)$schA = Filter (%a. a:act A)$tr &
-  Filter (%a. a:ext B)$schB = Filter (%a. a:act B)$tr &
-  LastActExtsch A schA & LastActExtsch B schB
-  --> Filter (%a. a:act B)$(mksch A B$tr$schA$schB) = schB"
-apply (intro strip)
-apply (rule seq.take_lemma)
-apply (rule mp)
-prefer 2 apply assumption
-back back back back
-apply (rule_tac x = "schA" in spec)
-apply (rule_tac x = "schB" in spec)
-apply (rule_tac x = "tr" in spec)
-apply (tactic "thin_tac' @{context} 5 1")
-apply (rule nat_less_induct)
-apply (rule allI)+
-apply (rename_tac tr schB schA)
-apply (intro strip)
-apply (erule conjE)+
+lemma FilterBmksch_is_schB:
+  "compatible A B \<Longrightarrow> compatible B A \<Longrightarrow> is_asig (asig_of A) \<Longrightarrow> is_asig (asig_of B) \<Longrightarrow>
+    Forall (\<lambda>x. x \<in> ext (A \<parallel> B)) tr \<and>
+    Forall (\<lambda>x. x \<in> act A) schA \<and>
+    Forall (\<lambda>x. x \<in> act B) schB \<and>
+    Filter (\<lambda>a. a \<in> ext A) $ schA = Filter (\<lambda>a. a \<in> act A) $ tr \<and>
+    Filter (\<lambda>a. a \<in> ext B) $ schB = Filter (\<lambda>a. a \<in> act B) $ tr \<and>
+    LastActExtsch A schA \<and> LastActExtsch B schB
+    \<longrightarrow> Filter (\<lambda>a. a \<in> act B) $ (mksch A B $ tr $ schA $ schB) = schB"
+  apply (intro strip)
+  apply (rule seq.take_lemma)
+  apply (rule mp)
+  prefer 2 apply assumption
+  back back back back
+  apply (rule_tac x = "schA" in spec)
+  apply (rule_tac x = "schB" in spec)
+  apply (rule_tac x = "tr" in spec)
+  apply (tactic "thin_tac' @{context} 5 1")
+  apply (rule nat_less_induct)
+  apply (rule allI)+
+  apply (rename_tac tr schB schA)
+  apply (intro strip)
+  apply (erule conjE)+
 
-apply (case_tac "Forall (%x. x:act A & x~:act B) tr")
+  apply (case_tac "Forall (\<lambda>x. x \<in> act A \<and> x \<notin> act B) tr")
 
-apply (rule seq_take_lemma [THEN iffD2, THEN spec])
-apply (tactic "thin_tac' @{context} 5 1")
+  apply (rule seq_take_lemma [THEN iffD2, THEN spec])
+  apply (tactic "thin_tac' @{context} 5 1")
 
-apply (case_tac "Finite tr")
+  apply (case_tac "Finite tr")
 
-(* both sides of this equation are nil *)
-apply (subgoal_tac "schB=nil")
-apply (simp (no_asm_simp))
-(* first side: mksch = nil *)
-apply (auto intro!: ForallQFilterPnil ForallAnBmksch FiniteL_mksch)[1]
-(* second side: schA = nil *)
-apply (erule_tac A = "B" in LastActExtimplnil)
-apply (simp (no_asm_simp))
-apply (erule_tac Q = "%x. x:act A & x~:act B" in ForallQFilterPnil)
-apply assumption
-apply fast
+  text \<open>both sides of this equation are \<open>nil\<close>\<close>
+  apply (subgoal_tac "schB = nil")
+  apply simp
+  text \<open>first side: \<open>mksch = nil\<close>\<close>
+  apply (auto intro!: ForallQFilterPnil ForallAnBmksch FiniteL_mksch)[1]
+  text \<open>second side: \<open>schA = nil\<close>\<close>
+  apply (erule_tac A = "B" in LastActExtimplnil)
+  apply (simp (no_asm_simp))
+  apply (erule_tac Q = "\<lambda>x. x \<in> act A \<and> x \<notin> act B" in ForallQFilterPnil)
+  apply assumption
+  apply fast
 
-(* case ~ Finite tr *)
+  text \<open>case \<open>\<not> Finite tr\<close>\<close>
 
-(* both sides of this equation are UU *)
-apply (subgoal_tac "schB=UU")
-apply (simp (no_asm_simp))
-(* first side: mksch = UU *)
-apply (force intro!: ForallQFilterPUU finiteR_mksch' ForallAnBmksch)
-(* schA = UU *)
-apply (erule_tac A = "B" in LastActExtimplUU)
-apply (simp (no_asm_simp))
-apply (erule_tac Q = "%x. x:act A & x~:act B" in ForallQFilterPUU)
-apply assumption
-apply fast
+  text \<open>both sides of this equation are \<open>UU\<close>\<close>
+  apply (subgoal_tac "schB = UU")
+  apply simp
+  text \<open>first side: \<open>mksch = UU\<close>\<close>
+  apply (force intro!: ForallQFilterPUU finiteR_mksch' ForallAnBmksch)
+  text \<open>\<open>schA = UU\<close>\<close>
+  apply (erule_tac A = "B" in LastActExtimplUU)
+  apply simp
+  apply (erule_tac Q = "\<lambda>x. x \<in> act A \<and> x \<notin> act B" in ForallQFilterPUU)
+  apply assumption
+  apply fast
 
-(* case" ~ Forall (%x.x:act B & x~:act A) s" *)
+  text \<open>case \<open>\<not> Forall (\<lambda>x. x \<in> act B \<and> x \<notin> act A) s\<close>\<close>
 
-apply (drule divide_Seq3)
+  apply (drule divide_Seq3)
 
-apply (erule exE)+
-apply (erule conjE)+
-apply hypsubst
+  apply (erule exE)+
+  apply (erule conjE)+
+  apply hypsubst
 
-(* bring in lemma reduceB_mksch *)
-apply (frule_tac y = "schB" and x = "schA" and A = "A" and B = "B" in reduceB_mksch)
-apply assumption+
-apply (erule exE)+
-apply (erule conjE)+
+  text \<open>bring in lemma \<open>reduceB_mksch\<close>\<close>
+  apply (frule_tac y = "schB" and x = "schA" and A = "A" and B = "B" in reduceB_mksch)
+  apply assumption+
+  apply (erule exE)+
+  apply (erule conjE)+
 
-(* use reduceB_mksch to rewrite conclusion *)
-apply hypsubst
-apply simp
+  text \<open>use \<open>reduceB_mksch\<close> to rewrite conclusion\<close>
+  apply hypsubst
+  apply simp
 
-(* eliminate the A-only prefix *)
+  text \<open>eliminate the \<open>A\<close>-only prefix\<close>
 
-apply (subgoal_tac "(Filter (%a. a :act B) $x1) = nil")
-apply (erule_tac [2] ForallQFilterPnil)
-prefer 2 apply (assumption)
-prefer 2 apply (fast)
+  apply (subgoal_tac "(Filter (\<lambda>a. a \<in> act B) $ x1) = nil")
+  apply (erule_tac [2] ForallQFilterPnil)
+  prefer 2 apply (assumption)
+  prefer 2 apply (fast)
 
-(* Now real recursive step follows (in x) *)
+  text \<open>Now real recursive step follows (in \<open>x\<close>)\<close>
 
-apply simp
-apply (case_tac "x:act B")
-apply (case_tac "x~:act A")
-apply (rotate_tac -2)
-apply simp
+  apply simp
+  apply (case_tac "x \<in> act B")
+  apply (case_tac "x \<notin> act A")
+  apply (rotate_tac -2)
+  apply simp
 
-apply (subgoal_tac "Filter (%a. a:act B & a:ext A) $x1=nil")
-apply (rotate_tac -1)
-apply simp
-(* eliminate introduced subgoal 2 *)
-apply (erule_tac [2] ForallQFilterPnil)
-prefer 2 apply (assumption)
-prefer 2 apply (fast)
+  apply (subgoal_tac "Filter (\<lambda>a. a \<in> act B \<and> a \<in> ext A) $ x1 = nil")
+  apply (rotate_tac -1)
+  apply simp
+  text \<open>eliminate introduced subgoal 2\<close>
+  apply (erule_tac [2] ForallQFilterPnil)
+  prefer 2 apply assumption
+  prefer 2 apply fast
 
-(* bring in divide Seq for s *)
-apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
-apply (erule conjE)+
+  text \<open>bring in \<open>divide_Seq\<close> for \<open>s\<close>\<close>
+  apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
+  apply (erule conjE)+
 
-(* subst divide_Seq in conclusion, but only at the rightmost occurrence *)
-apply (rule_tac t = "schB" in ssubst)
-back
-back
-back
-apply assumption
+  text \<open>subst \<open>divide_Seq\<close> in conclusion, but only at the rightmost occurrence\<close>
+  apply (rule_tac t = "schB" in ssubst)
+  back
+  back
+  back
+  apply assumption
 
-(* reduce trace_takes from n to strictly smaller k *)
-apply (rule take_reduction)
+  text \<open>reduce \<open>trace_takes\<close> from \<open>n\<close> to strictly smaller \<open>k\<close>\<close>
+  apply (rule take_reduction)
 
-(* f B (tw iB) = tw ~eB *)
-apply (simp add: int_is_act not_ext_is_int_or_not_act)
-apply (rule refl)
-apply (simp add: int_is_act not_ext_is_int_or_not_act)
-apply (rotate_tac -11)
+  text \<open>\<open>f B (tw iB) = tw \<not> eB\<close>\<close>
+  apply (simp add: int_is_act not_ext_is_int_or_not_act)
+  apply (rule refl)
+  apply (simp add: int_is_act not_ext_is_int_or_not_act)
+  apply (rotate_tac -11)
 
-(* now conclusion fulfills induction hypothesis, but assumptions are not ready *)
+  text \<open>now conclusion fulfills induction hypothesis, but assumptions are not ready\<close>
 
-(* assumption schA *)
-apply (simp add: ext_and_act)
-(* assumption schB *)
-apply (drule_tac x = "schB" and g = "Filter (%a. a:act B) $rs" in subst_lemma2)
-apply assumption
-apply (simp add: int_is_not_ext)
-(* assumptions concerning LastActExtsch, cannot be rewritten, as LastActExtsmalli are looping  *)
-apply (drule_tac sch = "schB" and P = "%a. a:int B" in LastActExtsmall1)
-apply (frule_tac ?sch1.0 = "x1" in LastActExtsmall2)
-apply assumption
+  text \<open>assumption \<open>schA\<close>\<close>
+  apply (simp add: ext_and_act)
+  text \<open>assumption \<open>schB\<close>\<close>
+  apply (drule_tac x = "schB" and g = "Filter (\<lambda>a. a \<in> act B) $ rs" in subst_lemma2)
+  apply assumption
+  apply (simp add: int_is_not_ext)
+  text \<open>assumptions concerning \<open>LastActExtsch\<close>, cannot be rewritten, as \<open>LastActExtsmalli\<close> are looping\<close>
+  apply (drule_tac sch = "schB" and P = "\<lambda>a. a \<in> int B" in LastActExtsmall1)
+  apply (frule_tac ?sch1.0 = "x1" in LastActExtsmall2)
+  apply assumption
 
-(* assumption Forall schB *)
-apply (drule_tac s = "schB" and P = "Forall (%x. x:act B) " in subst)
-apply assumption
-apply (simp add: int_is_act)
+  text \<open>assumption \<open>Forall schB\<close>\<close>
+  apply (drule_tac s = "schB" and P = "Forall (\<lambda>x. x \<in> act B)" in subst)
+  apply assumption
+  apply (simp add: int_is_act)
 
-(* case x:actions(asig_of A) & x: actions(asig_of B) *)
+  text \<open>case \<open>x \<in> actions (asig_of A) \<and> x \<in> actions (asig_of B)\<close>\<close>
 
-apply (rotate_tac -2)
-apply simp
+  apply (rotate_tac -2)
+  apply simp
 
-apply (subgoal_tac "Filter (%a. a:act B & a:ext A) $x1=nil")
-apply (rotate_tac -1)
-apply simp
-(* eliminate introduced subgoal 2 *)
-apply (erule_tac [2] ForallQFilterPnil)
-prefer 2 apply (assumption)
-prefer 2 apply (fast)
+  apply (subgoal_tac "Filter (\<lambda>a. a \<in> act B \<and> a \<in> ext A) $ x1 = nil")
+  apply (rotate_tac -1)
+  apply simp
+  text \<open>eliminate introduced subgoal 2\<close>
+  apply (erule_tac [2] ForallQFilterPnil)
+  prefer 2 apply assumption
+  prefer 2 apply fast
 
-(* bring in divide Seq for s *)
-apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
-apply (erule conjE)+
+  text \<open>bring in \<open>divide_Seq\<close> for \<open>s\<close>\<close>
+  apply (frule sym [THEN eq_imp_below, THEN divide_Seq])
+  apply (erule conjE)+
 
-(* subst divide_Seq in conclusion, but only at the rightmost occurrence *)
-apply (rule_tac t = "schB" in ssubst)
-back
-back
-back
-apply assumption
+  text \<open>subst \<open>divide_Seq\<close> in conclusion, but only at the rightmost occurrence\<close>
+  apply (rule_tac t = "schB" in ssubst)
+  back
+  back
+  back
+  apply assumption
 
-(* f B (tw iB) = tw ~eB *)
-apply (simp add: int_is_act not_ext_is_int_or_not_act)
+  text \<open>\<open>f B (tw iB) = tw \<not> eB\<close>\<close>
+  apply (simp add: int_is_act not_ext_is_int_or_not_act)
 
-(* rewrite assumption forall and schB *)
-apply (rotate_tac 13)
-apply (simp add: ext_and_act)
+  text \<open>rewrite assumption forall and \<open>schB\<close>\<close>
+  apply (rotate_tac 13)
+  apply (simp add: ext_and_act)
 
-(* divide_Seq for schB2 *)
-apply (frule_tac y = "x2" in sym [THEN eq_imp_below, THEN divide_Seq])
-apply (erule conjE)+
-(* assumption schA *)
-apply (drule_tac x = "schB" and g = "Filter (%a. a:act B) $rs" in subst_lemma2)
-apply assumption
-apply (simp add: int_is_not_ext)
+  text \<open>\<open>divide_Seq\<close> for \<open>schB2\<close>\<close>
+  apply (frule_tac y = "x2" in sym [THEN eq_imp_below, THEN divide_Seq])
+  apply (erule conjE)+
+  text \<open>assumption \<open>schA\<close>\<close>
+  apply (drule_tac x = "schB" and g = "Filter (\<lambda>a. a \<in> act B) $ rs" in subst_lemma2)
+  apply assumption
+  apply (simp add: int_is_not_ext)
 
-(* f B (tw iA schA2) = nil *)
-apply (simp add: int_is_not_ext not_ext_is_int_or_not_act intA_is_not_actB)
+  text \<open>\<open>f B (tw iA schA2) = nil\<close>\<close>
+  apply (simp add: int_is_not_ext not_ext_is_int_or_not_act intA_is_not_actB)
 
 
-(* reduce trace_takes from n to strictly smaller k *)
-apply (rule take_reduction)
-apply (rule refl)
-apply (rule refl)
+  text \<open>reduce \<open>trace_takes from \<open>n\<close> to strictly smaller \<open>k\<close>\<close>\<close>
+  apply (rule take_reduction)
+  apply (rule refl)
+  apply (rule refl)
 
-(* now conclusion fulfills induction hypothesis, but assumptions are not all ready *)
+  text \<open>now conclusion fulfills induction hypothesis, but assumptions are not all ready\<close>
 
-(* assumption schA *)
-apply (drule_tac x = "x2" and g = "Filter (%a. a:act A) $rs" in subst_lemma2)
-apply assumption
-apply (simp add: intA_is_not_actB int_is_not_ext)
+  text \<open>assumption \<open>schA\<close>\<close>
+  apply (drule_tac x = "x2" and g = "Filter (\<lambda>a. a \<in> act A) $ rs" in subst_lemma2)
+  apply assumption
+  apply (simp add: intA_is_not_actB int_is_not_ext)
 
-(* conclusions concerning LastActExtsch, cannot be rewritten, as LastActExtsmalli are looping  *)
-apply (drule_tac sch = "schB" and P = "%a. a:int B" in LastActExtsmall1)
-apply (frule_tac ?sch1.0 = "x1" in LastActExtsmall2)
-apply assumption
-apply (drule_tac sch = "x2" and P = "%a. a:int A" in LastActExtsmall1)
+  text \<open>conclusions concerning \<open>LastActExtsch\<close>, cannot be rewritten, as \<open>LastActExtsmalli\<close> are looping\<close>
+  apply (drule_tac sch = "schB" and P = "%a. a:int B" in LastActExtsmall1)
+  apply (frule_tac ?sch1.0 = "x1" in LastActExtsmall2)
+  apply assumption
+  apply (drule_tac sch = "x2" and P = "%a. a:int A" in LastActExtsmall1)
 
-(* assumption Forall schA, and Forall schA are performed by ForallTL,ForallDropwhile *)
-apply (simp add: ForallTL ForallDropwhile)
+  text \<open>assumption \<open>Forall schA\<close>, and \<open>Forall schA\<close> are performed by \<open>ForallTL\<close>, \<open>ForallDropwhile\<close>\<close>
+  apply (simp add: ForallTL ForallDropwhile)
 
-(* case x~:B & x:A  *)
-(* cannot occur, as just this case has been scheduled out before as the B-only prefix *)
-apply (case_tac "x:act A")
-apply fast
+  text \<open>case \<open>x \<notin> B \<and> x \<in> A\<close>\<close>
+  text \<open>cannot occur, as just this case has been scheduled out before as the \<open>B\<close>-only prefix\<close>
+  apply (case_tac "x \<in> act A")
+  apply fast
 
-(* case x~:B & x~:A  *)
-(* cannot occur because of assumption: Forall (a:ext A | a:ext B) *)
-apply (rotate_tac -9)
-(* reduce forall assumption from tr to (x\<leadsto>rs) *)
-apply (simp add: externals_of_par)
-apply (fast intro!: ext_is_act)
-
-done
+  text \<open>case \<open>x \<notin> B \<and> x \<notin> A\<close>\<close>
+  text \<open>cannot occur because of assumption: \<open>Forall (a \<in> ext A \<or> a \<in> ext B)\<close>\<close>
+  apply (rotate_tac -9)
+  text \<open>reduce forall assumption from \<open>tr\<close> to \<open>x \<leadsto> rs\<close>\<close>
+  apply (simp add: externals_of_par)
+  apply (fast intro!: ext_is_act)
+  done
 
 
 subsection "COMPOSITIONALITY on TRACE Level -- Main Theorem"
 
-lemma compositionality_tr:
-"!! A B. [| is_trans_of A; is_trans_of B; compatible A B; compatible B A;
-            is_asig(asig_of A); is_asig(asig_of B)|]
-        ==>  (tr: traces(A\<parallel>B)) =
-             (Filter (%a. a:act A)$tr : traces A &
-              Filter (%a. a:act B)$tr : traces B &
-              Forall (%x. x:ext(A\<parallel>B)) tr)"
-
-apply (simp (no_asm) add: traces_def has_trace_def)
-apply auto
+theorem compositionality_tr:
+  "is_trans_of A \<Longrightarrow> is_trans_of B \<Longrightarrow> compatible A B \<Longrightarrow> compatible B A \<Longrightarrow>
+    is_asig (asig_of A) \<Longrightarrow> is_asig (asig_of B) \<Longrightarrow>
+    tr \<in> traces (A \<parallel> B) \<longleftrightarrow>
+      Filter (\<lambda>a. a \<in> act A) $ tr \<in> traces A \<and>
+      Filter (\<lambda>a. a \<in> act B) $ tr \<in> traces B \<and>
+      Forall (\<lambda>x. x \<in> ext(A \<parallel> B)) tr"
+  apply (simp add: traces_def has_trace_def)
+  apply auto
 
-(* ==> *)
-(* There is a schedule of A *)
-apply (rule_tac x = "Filter (%a. a:act A) $sch" in bexI)
-prefer 2
-apply (simp add: compositionality_sch)
-apply (simp add: compatibility_consequence1 externals_of_par ext1_ext2_is_not_act1)
-(* There is a schedule of B *)
-apply (rule_tac x = "Filter (%a. a:act B) $sch" in bexI)
-prefer 2
-apply (simp add: compositionality_sch)
-apply (simp add: compatibility_consequence2 externals_of_par ext1_ext2_is_not_act2)
-(* Traces of A\<parallel>B have only external actions from A or B *)
-apply (rule ForallPFilterP)
+  text \<open>\<open>\<Longrightarrow>\<close>\<close>
+  text \<open>There is a schedule of \<open>A\<close>\<close>
+  apply (rule_tac x = "Filter (\<lambda>a. a \<in> act A) $ sch" in bexI)
+  prefer 2
+  apply (simp add: compositionality_sch)
+  apply (simp add: compatibility_consequence1 externals_of_par ext1_ext2_is_not_act1)
+  text \<open>There is a schedule of \<open>B\<close>\<close>
+  apply (rule_tac x = "Filter (\<lambda>a. a \<in> act B) $ sch" in bexI)
+  prefer 2
+  apply (simp add: compositionality_sch)
+  apply (simp add: compatibility_consequence2 externals_of_par ext1_ext2_is_not_act2)
+  text \<open>Traces of \<open>A \<parallel> B\<close> have only external actions from \<open>A\<close> or \<open>B\<close>\<close>
+  apply (rule ForallPFilterP)
 
-(* <== *)
+  text \<open>\<open>\<Longleftarrow>\<close>\<close>
 
-(* replace schA and schB by Cut(schA) and Cut(schB) *)
-apply (drule exists_LastActExtsch)
-apply assumption
-apply (drule exists_LastActExtsch)
-apply assumption
-apply (erule exE)+
-apply (erule conjE)+
-(* Schedules of A(B) have only actions of A(B) *)
-apply (drule scheds_in_sig)
-apply assumption
-apply (drule scheds_in_sig)
-apply assumption
+  text \<open>replace \<open>schA\<close> and \<open>schB\<close> by \<open>Cut schA\<close> and \<open>Cut schB\<close>\<close>
+  apply (drule exists_LastActExtsch)
+  apply assumption
+  apply (drule exists_LastActExtsch)
+  apply assumption
+  apply (erule exE)+
+  apply (erule conjE)+
+  text \<open>Schedules of A(B) have only actions of A(B)\<close>
+  apply (drule scheds_in_sig)
+  apply assumption
+  apply (drule scheds_in_sig)
+  apply assumption
 
-apply (rename_tac h1 h2 schA schB)
-(* mksch is exactly the construction of trA\<parallel>B out of schA, schB, and the oracle tr,
-   we need here *)
-apply (rule_tac x = "mksch A B$tr$schA$schB" in bexI)
+  apply (rename_tac h1 h2 schA schB)
+  text \<open>\<open>mksch\<close> is exactly the construction of \<open>trA\<parallel>B\<close> out of \<open>schA\<close>, \<open>schB\<close>, and the oracle \<open>tr\<close>,
+     we need here\<close>
+  apply (rule_tac x = "mksch A B $ tr $ schA $ schB" in bexI)
 
-(* External actions of mksch are just the oracle *)
-apply (simp add: FilterA_mksch_is_tr)
+  text \<open>External actions of mksch are just the oracle\<close>
+  apply (simp add: FilterA_mksch_is_tr)
 
-(* mksch is a schedule -- use compositionality on sch-level *)
-apply (simp add: compositionality_sch)
-apply (simp add: FilterAmksch_is_schA FilterBmksch_is_schB)
-apply (erule ForallAorB_mksch)
-apply (erule ForallPForallQ)
-apply (erule ext_is_act)
-done
+  text \<open>\<open>mksch\<close> is a schedule -- use compositionality on sch-level\<close>
+  apply (simp add: compositionality_sch)
+  apply (simp add: FilterAmksch_is_schA FilterBmksch_is_schB)
+  apply (erule ForallAorB_mksch)
+  apply (erule ForallPForallQ)
+  apply (erule ext_is_act)
+  done
 
 
 
 subsection \<open>COMPOSITIONALITY on TRACE Level -- for Modules\<close>
 
 lemma compositionality_tr_modules:
-
-"!! A B. [| is_trans_of A; is_trans_of B; compatible A B; compatible B A;
-            is_asig(asig_of A); is_asig(asig_of B)|]
- ==> Traces (A\<parallel>B) = par_traces (Traces A) (Traces B)"
-
-apply (unfold Traces_def par_traces_def)
-apply (simp add: asig_of_par)
-apply (rule set_eqI)
-apply (simp add: compositionality_tr externals_of_par)
-done
+  "is_trans_of A \<Longrightarrow> is_trans_of B \<Longrightarrow> compatible A B \<Longrightarrow> compatible B A \<Longrightarrow>
+    is_asig(asig_of A) \<Longrightarrow> is_asig(asig_of B) \<Longrightarrow>
+    Traces (A\<parallel>B) = par_traces (Traces A) (Traces B)"
+  apply (unfold Traces_def par_traces_def)
+  apply (simp add: asig_of_par)
+  apply (rule set_eqI)
+  apply (simp add: compositionality_tr externals_of_par)
+  done
 
 
 declaration \<open>fn _ => Simplifier.map_ss (Simplifier.set_mksym Simplifier.default_mk_sym)\<close>
 
-
 end
diff -r ec2f0dad8b98 -r 7355fd313cf8 src/HOL/HOLCF/IOA/Deadlock.thy
--- a/src/HOL/HOLCF/IOA/Deadlock.thy	Tue Jan 12 20:05:53 2016 +0100
+++ b/src/HOL/HOLCF/IOA/Deadlock.thy	Tue Jan 12 23:40:33 2016 +0100
@@ -8,85 +8,91 @@
 imports RefCorrectness CompoScheds
 begin
 
-text \<open>input actions may always be added to a schedule\<close>
+text \<open>Input actions may always be added to a schedule.\<close>
 
 lemma scheds_input_enabled:
-  "[| Filter (%x. x:act A)$sch : schedules A; a:inp A; input_enabled A; Finite sch|]  
-          ==> Filter (%x. x:act A)$sch @@ a\<leadsto>nil : schedules A"
-apply (simp add: schedules_def has_schedule_def)
-apply auto
-apply (frule inp_is_act)
-apply (simp add: executions_def)
-apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
-apply (rename_tac s ex)
-apply (subgoal_tac "Finite ex")
-prefer 2
-apply (simp add: filter_act_def)
-defer
-apply (rule_tac [2] Map2Finite [THEN iffD1])
-apply (rule_tac [2] t = "Map fst$ex" in subst)
-prefer 2 apply (assumption)
-apply (erule_tac [2] FiniteFilter)
-(* subgoal 1 *)
-apply (frule exists_laststate)
-apply (erule allE)
-apply (erule exE)
-(* using input-enabledness *)
-apply (simp add: input_enabled_def)
-apply (erule conjE)+
-apply (erule_tac x = "a" in allE)
-apply simp
-apply (erule_tac x = "u" in allE)
-apply (erule exE)
-(* instantiate execution *)
-apply (rule_tac x = " (s,ex @@ (a,s2) \<leadsto>nil) " in exI)
-apply (simp add: filter_act_def MapConc)
-apply (erule_tac t = "u" in lemma_2_1)
-apply simp
-apply (rule sym)
-apply assumption
-done
+  "Filter (\<lambda>x. x \<in> act A) $ sch \<in> schedules A \<Longrightarrow> a \<in> inp A \<Longrightarrow> input_enabled A \<Longrightarrow> Finite sch
+    \<Longrightarrow> Filter (\<lambda>x. x \<in> act A) $ sch @@ a \<leadsto> nil \<in> schedules A"
+  apply (simp add: schedules_def has_schedule_def)
+  apply auto
+  apply (frule inp_is_act)
+  apply (simp add: executions_def)
+  apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+  apply (rename_tac s ex)
+  apply (subgoal_tac "Finite ex")
+  prefer 2
+  apply (simp add: filter_act_def)
+  defer
+  apply (rule_tac [2] Map2Finite [THEN iffD1])
+  apply (rule_tac [2] t = "Map fst $ ex" in subst)
+  prefer 2
+  apply assumption
+  apply (erule_tac [2] FiniteFilter)
+  text \<open>subgoal 1\<close>
+  apply (frule exists_laststate)
+  apply (erule allE)
+  apply (erule exE)
+  text \<open>using input-enabledness\<close>
+  apply (simp add: input_enabled_def)
+  apply (erule conjE)+
+  apply (erule_tac x = "a" in allE)
+  apply simp
+  apply (erule_tac x = "u" in allE)
+  apply (erule exE)
+  text \<open>instantiate execution\<close>
+  apply (rule_tac x = " (s, ex @@ (a, s2) \<leadsto> nil) " in exI)
+  apply (simp add: filter_act_def MapConc)
+  apply (erule_tac t = "u" in lemma_2_1)
+  apply simp
+  apply (rule sym)
+  apply assumption
+  done
 
 text \<open>
-               Deadlock freedom: component B cannot block an out or int action
-                                 of component A in every schedule.
-    Needs compositionality on schedule level, input-enabledness, compatibility
-                    and distributivity of is_exec_frag over @@
+  Deadlock freedom: component B cannot block an out or int action of component
+  A in every schedule.
+
+  Needs compositionality on schedule level, input-enabledness, compatibility
+  and distributivity of \<open>is_exec_frag\<close> over \<open>@@\<close>.
 \<close>
 
-declare split_if [split del]
-lemma IOA_deadlock_free: "[| a : local A; Finite sch; sch : schedules (A\<parallel>B);  
-             Filter (%x. x:act A)$(sch @@ a\<leadsto>nil) : schedules A; compatible A B; input_enabled B |]  
-           ==> (sch @@ a\<leadsto>nil) : schedules (A\<parallel>B)"
-apply (simp add: compositionality_sch locals_def)
-apply (rule conjI)
-(* a : act (A\<parallel>B) *)
-prefer 2
-apply (simp add: actions_of_par)
-apply (blast dest: int_is_act out_is_act)
-
-(* Filter B (sch@@[a]) : schedules B *)
+lemma IOA_deadlock_free:
+  assumes "a \<in> local A"
+    and "Finite sch"
+    and "sch \<in> schedules (A \<parallel> B)"
+    and "Filter (\<lambda>x. x \<in> act A) $ (sch @@ a \<leadsto> nil) \<in> schedules A"
+    and "compatible A B"
+    and "input_enabled B"
+  shows "(sch @@ a \<leadsto> nil) \<in> schedules (A \<parallel> B)"
+  supply split_if [split del]
+  apply (insert assms)
+  apply (simp add: compositionality_sch locals_def)
+  apply (rule conjI)
+  text \<open>\<open>a \<in> act (A \<parallel> B)\<close>\<close>
+  prefer 2
+  apply (simp add: actions_of_par)
+  apply (blast dest: int_is_act out_is_act)
 
-apply (case_tac "a:int A")
-apply (drule intA_is_not_actB)
-apply (assumption) (* --> a~:act B *)
-apply simp
+  text \<open>\<open>Filter B (sch @@ [a]) \<in> schedules B\<close>\<close>
+  apply (case_tac "a \<in> int A")
+  apply (drule intA_is_not_actB)
+  apply (assumption) (* \<longrightarrow> a \<notin> act B *)
+  apply simp
 
-(* case a~:int A , i.e. a:out A *)
-apply (case_tac "a~:act B")
-apply simp
-(* case a:act B *)
-apply simp
-apply (subgoal_tac "a:out A")
-prefer 2 apply (blast)
-apply (drule outAactB_is_inpB)
-apply assumption
-apply assumption
-apply (rule scheds_input_enabled)
-apply simp
-apply assumption+
-done
-
-declare split_if [split]
+  text \<open>case \<open>a \<notin> int A\<close>, i.e. \<open>a \<in> out A\<close>\<close>
+  apply (case_tac "a \<notin> act B")
+  apply simp
+  text \<open>case \<open>a \<in> act B\<close>\<close>
+  apply simp
+  apply (subgoal_tac "a \<in> out A")
+  prefer 2
+  apply blast
+  apply (drule outAactB_is_inpB)
+  apply assumption
+  apply assumption
+  apply (rule scheds_input_enabled)
+  apply simp
+  apply assumption+
+  done
 
 end
diff -r ec2f0dad8b98 -r 7355fd313cf8 src/HOL/HOLCF/IOA/RefCorrectness.thy
--- a/src/HOL/HOLCF/IOA/RefCorrectness.thy	Tue Jan 12 20:05:53 2016 +0100
+++ b/src/HOL/HOLCF/IOA/RefCorrectness.thy	Tue Jan 12 23:40:33 2016 +0100
@@ -8,363 +8,349 @@
 imports RefMappings
 begin
 
-definition
-  corresp_exC :: "('a,'s2)ioa => ('s1 => 's2) => ('a,'s1)pairs
-                   -> ('s1 => ('a,'s2)pairs)" where
-  "corresp_exC A f = (fix$(LAM h ex. (%s. case ex of
-      nil =>  nil
-    | x##xs => (flift1 (%pr. (@cex. move A cex (f s) (fst pr) (f (snd pr)))
-                              @@ ((h$xs) (snd pr)))
-                        $x) )))"
-definition
-  corresp_ex :: "('a,'s2)ioa => ('s1 => 's2) =>
-                  ('a,'s1)execution => ('a,'s2)execution" where
-  "corresp_ex A f ex = (f (fst ex),(corresp_exC A f$(snd ex)) (fst ex))"
+definition corresp_exC ::
+    "('a, 's2) ioa \<Rightarrow> ('s1 \<Rightarrow> 's2) \<Rightarrow> ('a, 's1) pairs \<rightarrow> ('s1 \<Rightarrow> ('a, 's2) pairs)"
+  where "corresp_exC A f =
+    (fix $
+      (LAM h ex.
+        (\<lambda>s. case ex of
+          nil \<Rightarrow> nil
+        | x ## xs \<Rightarrow>
+            flift1 (\<lambda>pr.
+              (SOME cex. move A cex (f s) (fst pr) (f (snd pr))) @@ ((h $ xs) (snd pr))) $ x)))"
 
-definition
-  is_fair_ref_map :: "('s1 => 's2) => ('a,'s1)ioa => ('a,'s2)ioa => bool" where
-  "is_fair_ref_map f C A =
-      (is_ref_map f C A &
-       (! ex : executions(C). fair_ex C ex --> fair_ex A (corresp_ex A f ex)))"
+definition corresp_ex ::
+    "('a, 's2) ioa \<Rightarrow> ('s1 \<Rightarrow> 's2) \<Rightarrow> ('a, 's1) execution \<Rightarrow> ('a, 's2) execution"
+  where "corresp_ex A f ex = (f (fst ex), (corresp_exC A f $ (snd ex)) (fst ex))"
 
-(* Axioms for fair trace inclusion proof support, not for the correctness proof
-   of refinement mappings!
-   Note: Everything is superseded by LiveIOA.thy! *)
+definition is_fair_ref_map ::
+    "('s1 \<Rightarrow> 's2) \<Rightarrow> ('a, 's1) ioa \<Rightarrow> ('a, 's2) ioa \<Rightarrow> bool"
+  where "is_fair_ref_map f C A \<longleftrightarrow>
+    is_ref_map f C A \<and> (\<forall>ex \<in> executions C. fair_ex C ex \<longrightarrow> fair_ex A (corresp_ex A f ex))"
+
+text \<open>
+  Axioms for fair trace inclusion proof support, not for the correctness proof
+  of refinement mappings!
+
+  Note: Everything is superseded by @{file "LiveIOA.thy"}.
+\<close>
 
 axiomatization where
-corresp_laststate:
-  "Finite ex ==> laststate (corresp_ex A f (s,ex)) = f (laststate (s,ex))"
+  corresp_laststate:
+    "Finite ex \<Longrightarrow> laststate (corresp_ex A f (s, ex)) = f (laststate (s, ex))"
+
+axiomatization where
+  corresp_Finite: "Finite (snd (corresp_ex A f (s, ex))) = Finite ex"
 
 axiomatization where
-corresp_Finite:
-  "Finite (snd (corresp_ex A f (s,ex))) = Finite ex"
+  FromAtoC:
+    "fin_often (\<lambda>x. P (snd x)) (snd (corresp_ex A f (s, ex))) \<Longrightarrow>
+      fin_often (\<lambda>y. P (f (snd y))) ex"
 
 axiomatization where
-FromAtoC:
-  "fin_often (%x. P (snd x)) (snd (corresp_ex A f (s,ex))) ==> fin_often (%y. P (f (snd y))) ex"
-
-axiomatization where
-FromCtoA:
-  "inf_often (%y. P (fst y)) ex ==> inf_often (%x. P (fst x)) (snd (corresp_ex A f (s,ex)))"
+  FromCtoA:
+    "inf_often (\<lambda>y. P (fst y)) ex \<Longrightarrow>
+      inf_often (\<lambda>x. P (fst x)) (snd (corresp_ex A f (s,ex)))"
 
 
-(* Proof by case on inf W in ex: If so, ok. If not, only fin W in ex, ie there is
-   an index i from which on no W in ex. But W inf enabled, ie at least once after i
-   W is enabled. As W does not occur after i and W is enabling_persistent, W keeps
-   enabled until infinity, ie. indefinitely *)
-axiomatization where
-persistent:
-  "[|inf_often (%x. Enabled A W (snd x)) ex; en_persistent A W|]
-   ==> inf_often (%x. fst x :W) ex | fin_often (%x. ~Enabled A W (snd x)) ex"
+text \<open>
+  Proof by case on \<open>inf W\<close> in ex: If so, ok. If not, only \<open>fin W\<close> in ex, ie.
+  there is an index \<open>i\<close> from which on no \<open>W\<close> in ex. But \<open>W\<close> inf enabled, ie at
+  least once after \<open>i\<close> \<open>W\<close> is enabled. As \<open>W\<close> does not occur after \<open>i\<close> and \<open>W\<close>
+  is \<open>enabling_persistent\<close>, \<open>W\<close> keeps enabled until infinity, ie. indefinitely
+\<close>
 
 axiomatization where
-infpostcond:
-  "[| is_exec_frag A (s,ex); inf_often (%x. fst x:W) ex|]
-    ==> inf_often (% x. set_was_enabled A W (snd x)) ex"
+  persistent:
+    "inf_often (\<lambda>x. Enabled A W (snd x)) ex \<Longrightarrow> en_persistent A W \<Longrightarrow>
+      inf_often (\<lambda>x. fst x \<in> W) ex \<or> fin_often (\<lambda>x. \<not> Enabled A W (snd x)) ex"
+
+axiomatization where
+  infpostcond:
+    "is_exec_frag A (s,ex) \<Longrightarrow> inf_often (\<lambda>x. fst x \<in> W) ex \<Longrightarrow>
+      inf_often (\<lambda>x. set_was_enabled A W (snd x)) ex"
 
 
-subsection "corresp_ex"
+subsection \<open>\<open>corresp_ex\<close>\<close>
 
-lemma corresp_exC_unfold: "corresp_exC A f  = (LAM ex. (%s. case ex of
-       nil =>  nil
-     | x##xs => (flift1 (%pr. (@cex. move A cex (f s) (fst pr) (f (snd pr)))
-                               @@ ((corresp_exC A f $xs) (snd pr)))
-                         $x) ))"
-apply (rule trans)
-apply (rule fix_eq2)
-apply (simp only: corresp_exC_def)
-apply (rule beta_cfun)
-apply (simp add: flift1_def)
-done
+lemma corresp_exC_unfold:
+  "corresp_exC A f =
+    (LAM ex.
+      (\<lambda>s.
+        case ex of
+          nil \<Rightarrow> nil
+        | x ## xs \<Rightarrow>
+            (flift1 (\<lambda>pr.
+              (SOME cex. move A cex (f s) (fst pr) (f (snd pr))) @@
+              ((corresp_exC A f $ xs) (snd pr))) $ x)))"
+  apply (rule trans)
+  apply (rule fix_eq2)
+  apply (simp only: corresp_exC_def)
+  apply (rule beta_cfun)
+  apply (simp add: flift1_def)
+  done
 
-lemma corresp_exC_UU: "(corresp_exC A f$UU) s=UU"
-apply (subst corresp_exC_unfold)
-apply simp
-done
+lemma corresp_exC_UU: "(corresp_exC A f $ UU) s = UU"
+  apply (subst corresp_exC_unfold)
+  apply simp
+  done
 
-lemma corresp_exC_nil: "(corresp_exC A f$nil) s = nil"
-apply (subst corresp_exC_unfold)
-apply simp
-done
+lemma corresp_exC_nil: "(corresp_exC A f $ nil) s = nil"
+  apply (subst corresp_exC_unfold)
+  apply simp
+  done
 
-lemma corresp_exC_cons: "(corresp_exC A f$(at\<leadsto>xs)) s =
-           (@cex. move A cex (f s) (fst at) (f (snd at)))
-           @@ ((corresp_exC A f$xs) (snd at))"
-apply (rule trans)
-apply (subst corresp_exC_unfold)
-apply (simp add: Consq_def flift1_def)
-apply simp
-done
-
+lemma corresp_exC_cons:
+  "(corresp_exC A f $ (at \<leadsto> xs)) s =
+     (SOME cex. move A cex (f s) (fst at) (f (snd at))) @@
+     ((corresp_exC A f $ xs) (snd at))"
+  apply (rule trans)
+  apply (subst corresp_exC_unfold)
+  apply (simp add: Consq_def flift1_def)
+  apply simp
+  done
 
 declare corresp_exC_UU [simp] corresp_exC_nil [simp] corresp_exC_cons [simp]
 
 
-
-subsection "properties of move"
+subsection \<open>Properties of move\<close>
 
 lemma move_is_move:
-   "[|is_ref_map f C A; reachable C s; (s,a,t):trans_of C|] ==>
-      move A (@x. move A x (f s) a (f t)) (f s) a (f t)"
-apply (unfold is_ref_map_def)
-apply (subgoal_tac "? ex. move A ex (f s) a (f t) ")
-prefer 2
-apply simp
-apply (erule exE)
-apply (rule someI)
-apply assumption
-done
+  "is_ref_map f C A \<Longrightarrow> reachable C s \<Longrightarrow> (s, a, t) \<in> trans_of C \<Longrightarrow>
+    move A (SOME x. move A x (f s) a (f t)) (f s) a (f t)"
+  apply (unfold is_ref_map_def)
+  apply (subgoal_tac "\<exists>ex. move A ex (f s) a (f t) ")
+  prefer 2
+  apply simp
+  apply (erule exE)
+  apply (rule someI)
+  apply assumption
+  done
 
 lemma move_subprop1:
-   "[|is_ref_map f C A; reachable C s; (s,a,t):trans_of C|] ==>
-     is_exec_frag A (f s,@x. move A x (f s) a (f t))"
-apply (cut_tac move_is_move)
-defer
-apply assumption+
-apply (simp add: move_def)
-done
+  "is_ref_map f C A \<Longrightarrow> reachable C s \<Longrightarrow> (s, a, t) \<in> trans_of C \<Longrightarrow>
+    is_exec_frag A (f s, SOME x. move A x (f s) a (f t))"
+  apply (cut_tac move_is_move)
+  defer
+  apply assumption+
+  apply (simp add: move_def)
+  done
 
 lemma move_subprop2:
-   "[|is_ref_map f C A; reachable C s; (s,a,t):trans_of C|] ==>
-     Finite ((@x. move A x (f s) a (f t)))"
-apply (cut_tac move_is_move)
-defer
-apply assumption+
-apply (simp add: move_def)
-done
+  "is_ref_map f C A \<Longrightarrow> reachable C s \<Longrightarrow> (s, a, t) \<in> trans_of C \<Longrightarrow>
+    Finite ((SOME x. move A x (f s) a (f t)))"
+  apply (cut_tac move_is_move)
+  defer
+  apply assumption+
+  apply (simp add: move_def)
+  done
 
 lemma move_subprop3:
-   "[|is_ref_map f C A; reachable C s; (s,a,t):trans_of C|] ==>
+  "is_ref_map f C A \<Longrightarrow> reachable C s \<Longrightarrow> (s, a, t) \<in> trans_of C \<Longrightarrow>
      laststate (f s,@x. move A x (f s) a (f t)) = (f t)"
-apply (cut_tac move_is_move)
-defer
-apply assumption+
-apply (simp add: move_def)
-done
+  apply (cut_tac move_is_move)
+  defer
+  apply assumption+
+  apply (simp add: move_def)
+  done
 
 lemma move_subprop4:
-   "[|is_ref_map f C A; reachable C s; (s,a,t):trans_of C|] ==>
-      mk_trace A$((@x. move A x (f s) a (f t))) =
-        (if a:ext A then a\<leadsto>nil else nil)"
-apply (cut_tac move_is_move)
-defer
-apply assumption+
-apply (simp add: move_def)
-done
+  "is_ref_map f C A \<Longrightarrow> reachable C s \<Longrightarrow> (s, a, t) \<in> trans_of C \<Longrightarrow>
+    mk_trace A $ ((SOME x. move A x (f s) a (f t))) =
+      (if a \<in> ext A then a \<leadsto> nil else nil)"
+  apply (cut_tac move_is_move)
+  defer
+  apply assumption+
+  apply (simp add: move_def)
+  done
 
 
-(* ------------------------------------------------------------------ *)
-(*                   The following lemmata contribute to              *)
-(*                 TRACE INCLUSION Part 1: Traces coincide            *)
-(* ------------------------------------------------------------------ *)
+subsection \<open>TRACE INCLUSION Part 1: Traces coincide\<close>
 
-section "Lemmata for <=="
+subsubsection \<open>Lemmata for \<open>\<Longleftarrow>\<close>\<close>
 
-(* --------------------------------------------------- *)
-(*   Lemma 1.1: Distribution of mk_trace and @@        *)
-(* --------------------------------------------------- *)
+text \<open>Lemma 1.1: Distribution of \<open>mk_trace\<close> and \<open>@@\<close>\<close>
 
-lemma mk_traceConc: "mk_trace C$(ex1 @@ ex2)= (mk_trace C$ex1) @@ (mk_trace C$ex2)"
-apply (simp add: mk_trace_def filter_act_def MapConc)
-done
+lemma mk_traceConc:
+  "mk_trace C $ (ex1 @@ ex2) = (mk_trace C $ ex1) @@ (mk_trace C $ ex2)"
+  by (simp add: mk_trace_def filter_act_def MapConc)
 
 
-
-(* ------------------------------------------------------
-                 Lemma 1 :Traces coincide
-   ------------------------------------------------------- *)
-declare split_if [split del]
+text \<open>Lemma 1 : Traces coincide\<close>
 
 lemma lemma_1:
-  "[|is_ref_map f C A; ext C = ext A|] ==>
-         !s. reachable C s & is_exec_frag C (s,xs) -->
-             mk_trace C$xs = mk_trace A$(snd (corresp_ex A f (s,xs)))"
-apply (unfold corresp_ex_def)
-apply (tactic \<open>pair_induct_tac @{context} "xs" [@{thm is_exec_frag_def}] 1\<close>)
-(* cons case *)
-apply (auto simp add: mk_traceConc)
-apply (frule reachable.reachable_n)
-apply assumption
-apply (auto simp add: move_subprop4 split add: split_if) 
-done
-
-declare split_if [split]
-
-(* ------------------------------------------------------------------ *)
-(*                   The following lemmata contribute to              *)
-(*              TRACE INCLUSION Part 2: corresp_ex is execution       *)
-(* ------------------------------------------------------------------ *)
-
-section "Lemmata for ==>"
-
-(* -------------------------------------------------- *)
-(*                   Lemma 2.1                        *)
-(* -------------------------------------------------- *)
-
-lemma lemma_2_1 [rule_format (no_asm)]:
-"Finite xs -->
- (!s .is_exec_frag A (s,xs) & is_exec_frag A (t,ys) &
-      t = laststate (s,xs)
-  --> is_exec_frag A (s,xs @@ ys))"
-
-apply (rule impI)
-apply (tactic \<open>Seq_Finite_induct_tac @{context} 1\<close>)
-(* main case *)
-apply (auto simp add: split_paired_all)
-done
+  "is_ref_map f C A \<Longrightarrow> ext C = ext A \<Longrightarrow>
+    \<forall>s. reachable C s \<and> is_exec_frag C (s, xs) \<longrightarrow>
+      mk_trace C $ xs = mk_trace A $ (snd (corresp_ex A f (s, xs)))"
+  supply split_if [split del]
+  apply (unfold corresp_ex_def)
+  apply (tactic \<open>pair_induct_tac @{context} "xs" [@{thm is_exec_frag_def}] 1\<close>)
+  text \<open>cons case\<close>
+  apply (auto simp add: mk_traceConc)
+  apply (frule reachable.reachable_n)
+  apply assumption
+  apply (auto simp add: move_subprop4 split add: split_if)
+  done
 
 
-(* ----------------------------------------------------------- *)
-(*               Lemma 2 : corresp_ex is execution             *)
-(* ----------------------------------------------------------- *)
+subsection \<open>TRACE INCLUSION Part 2: corresp_ex is execution\<close>
+
+subsubsection \<open>Lemmata for \<open>==>\<close>\<close>
+
+text \<open>Lemma 2.1\<close>
+
+lemma lemma_2_1 [rule_format]:
+  "Finite xs \<longrightarrow>
+    (\<forall>s. is_exec_frag A (s, xs) \<and> is_exec_frag A (t, ys) \<and>
+      t = laststate (s, xs) \<longrightarrow> is_exec_frag A (s, xs @@ ys))"
+  apply (rule impI)
+  apply (tactic \<open>Seq_Finite_induct_tac @{context} 1\<close>)
+  text \<open>main case\<close>
+  apply (auto simp add: split_paired_all)
+  done
 
 
+text \<open>Lemma 2 : \<open>corresp_ex\<close> is execution\<close>
 
 lemma lemma_2:
- "[| is_ref_map f C A |] ==>
-  !s. reachable C s & is_exec_frag C (s,xs)
-  --> is_exec_frag A (corresp_ex A f (s,xs))"
+  "is_ref_map f C A \<Longrightarrow>
+    \<forall>s. reachable C s \<and> is_exec_frag C (s, xs) \<longrightarrow>
+      is_exec_frag A (corresp_ex A f (s, xs))"
+  apply (unfold corresp_ex_def)
 
-apply (unfold corresp_ex_def)
+  apply simp
+  apply (tactic \<open>pair_induct_tac @{context} "xs" [@{thm is_exec_frag_def}] 1\<close>)
 
-apply simp
-apply (tactic \<open>pair_induct_tac @{context} "xs" [@{thm is_exec_frag_def}] 1\<close>)
-(* main case *)
-apply auto
-apply (rule_tac t = "f x2" in lemma_2_1)
+  text \<open>main case\<close>
+  apply auto
+  apply (rule_tac t = "f x2" in lemma_2_1)
 
-(* Finite *)
-apply (erule move_subprop2)
-apply assumption+
-apply (rule conjI)
+  text \<open>\<open>Finite\<close>\<close>
+  apply (erule move_subprop2)
+  apply assumption+
+  apply (rule conjI)
 
-(* is_exec_frag *)
-apply (erule move_subprop1)
-apply assumption+
-apply (rule conjI)
+  text \<open>\<open>is_exec_frag\<close>\<close>
+  apply (erule move_subprop1)
+  apply assumption+
+  apply (rule conjI)
 
-(* Induction hypothesis  *)
-(* reachable_n looping, therefore apply it manually *)
-apply (erule_tac x = "x2" in allE)
-apply simp
-apply (frule reachable.reachable_n)
-apply assumption
-apply simp
-(* laststate *)
-apply (erule move_subprop3 [symmetric])
-apply assumption+
-done
+  text \<open>Induction hypothesis\<close>
+  text \<open>\<open>reachable_n\<close> looping, therefore apply it manually\<close>
+  apply (erule_tac x = "x2" in allE)
+  apply simp
+  apply (frule reachable.reachable_n)
+  apply assumption
+  apply simp
+
+  text \<open>\<open>laststate\<close>\<close>
+  apply (erule move_subprop3 [symmetric])
+  apply assumption+
+  done
 
 
-subsection "Main Theorem: TRACE - INCLUSION"
+subsection \<open>Main Theorem: TRACE -- INCLUSION\<close>
 
-lemma trace_inclusion:
-  "[| ext C = ext A; is_ref_map f C A |]
-           ==> traces C <= traces A"
+theorem trace_inclusion:
+  "ext C = ext A \<Longrightarrow> is_ref_map f C A \<Longrightarrow> traces C \<subseteq> traces A"
 
   apply (unfold traces_def)
 
-  apply (simp (no_asm) add: has_trace_def2)
+  apply (simp add: has_trace_def2)
   apply auto
 
-  (* give execution of abstract automata *)
+  text \<open>give execution of abstract automata\<close>
   apply (rule_tac x = "corresp_ex A f ex" in bexI)
 
-  (* Traces coincide, Lemma 1 *)
+  text \<open>Traces coincide, Lemma 1\<close>
   apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
   apply (erule lemma_1 [THEN spec, THEN mp])
   apply assumption+
   apply (simp add: executions_def reachable.reachable_0)
 
-  (* corresp_ex is execution, Lemma 2 *)
+  text \<open>\<open>corresp_ex\<close> is execution, Lemma 2\<close>
   apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
   apply (simp add: executions_def)
-  (* start state *)
+  text \<open>start state\<close>
   apply (rule conjI)
   apply (simp add: is_ref_map_def corresp_ex_def)
-  (* is-execution-fragment *)
+  text \<open>\<open>is_execution_fragment\<close>\<close>
   apply (erule lemma_2 [THEN spec, THEN mp])
   apply (simp add: reachable.reachable_0)
   done
 
 
-subsection "Corollary:  FAIR TRACE - INCLUSION"
+subsection \<open>Corollary:  FAIR TRACE -- INCLUSION\<close>
 
 lemma fininf: "(~inf_often P s) = fin_often P s"
-apply (unfold fin_often_def)
-apply auto
-done
-
+  by (auto simp: fin_often_def)
 
-lemma WF_alt: "is_wfair A W (s,ex) =
-  (fin_often (%x. ~Enabled A W (snd x)) ex --> inf_often (%x. fst x :W) ex)"
-apply (simp add: is_wfair_def fin_often_def)
-apply auto
-done
+lemma WF_alt: "is_wfair A W (s, ex) =
+  (fin_often (\<lambda>x. \<not> Enabled A W (snd x)) ex \<longrightarrow> inf_often (\<lambda>x. fst x \<in> W) ex)"
+  by (auto simp add: is_wfair_def fin_often_def)
 
-lemma WF_persistent: "[|is_wfair A W (s,ex); inf_often (%x. Enabled A W (snd x)) ex;
-          en_persistent A W|]
-    ==> inf_often (%x. fst x :W) ex"
-apply (drule persistent)
-apply assumption
-apply (simp add: WF_alt)
-apply auto
-done
-
+lemma WF_persistent:
+  "is_wfair A W (s, ex) \<Longrightarrow> inf_often (\<lambda>x. Enabled A W (snd x)) ex \<Longrightarrow>
+    en_persistent A W \<Longrightarrow> inf_often (\<lambda>x. fst x \<in> W) ex"
+  apply (drule persistent)
+  apply assumption
+  apply (simp add: WF_alt)
+  apply auto
+  done
 
-lemma fair_trace_inclusion: "!! C A.
-          [| is_ref_map f C A; ext C = ext A;
-          !! ex. [| ex:executions C; fair_ex C ex|] ==> fair_ex A (corresp_ex A f ex) |]
-          ==> fairtraces C <= fairtraces A"
-apply (simp (no_asm) add: fairtraces_def fairexecutions_def)
-apply auto
-apply (rule_tac x = "corresp_ex A f ex" in exI)
-apply auto
+lemma fair_trace_inclusion:
+  assumes "is_ref_map f C A"
+    and "ext C = ext A"
+    and "\<And>ex. ex \<in> executions C \<Longrightarrow> fair_ex C ex \<Longrightarrow>
+      fair_ex A (corresp_ex A f ex)"
+  shows "fairtraces C \<subseteq> fairtraces A"
+  apply (insert assms)
+  apply (simp add: fairtraces_def fairexecutions_def)
+  apply auto
+  apply (rule_tac x = "corresp_ex A f ex" in exI)
+  apply auto
 
-  (* Traces coincide, Lemma 1 *)
+  text \<open>Traces coincide, Lemma 1\<close>
   apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
   apply (erule lemma_1 [THEN spec, THEN mp])
   apply assumption+
   apply (simp add: executions_def reachable.reachable_0)
 
-  (* corresp_ex is execution, Lemma 2 *)
+  text \<open>\<open>corresp_ex\<close> is execution, Lemma 2\<close>
   apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
   apply (simp add: executions_def)
-  (* start state *)
+  text \<open>start state\<close>
   apply (rule conjI)
   apply (simp add: is_ref_map_def corresp_ex_def)
-  (* is-execution-fragment *)
+  text \<open>\<open>is_execution_fragment\<close>\<close>
   apply (erule lemma_2 [THEN spec, THEN mp])
   apply (simp add: reachable.reachable_0)
-
-done
+  done
 
-lemma fair_trace_inclusion2: "!! C A.
-          [| inp(C) = inp(A); out(C)=out(A);
-             is_fair_ref_map f C A |]
-          ==> fair_implements C A"
-apply (simp add: is_fair_ref_map_def fair_implements_def fairtraces_def fairexecutions_def)
-apply auto
-apply (rule_tac x = "corresp_ex A f ex" in exI)
-apply auto
+lemma fair_trace_inclusion2:
+  "inp C = inp A \<Longrightarrow> out C = out A \<Longrightarrow> is_fair_ref_map f C A \<Longrightarrow>
+    fair_implements C A"
+  apply (simp add: is_fair_ref_map_def fair_implements_def fairtraces_def fairexecutions_def)
+  apply auto
+  apply (rule_tac x = "corresp_ex A f ex" in exI)
+  apply auto
 
-  (* Traces coincide, Lemma 1 *)
+  text \<open>Traces coincide, Lemma 1\<close>
   apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
   apply (erule lemma_1 [THEN spec, THEN mp])
   apply (simp (no_asm) add: externals_def)
   apply (auto)[1]
   apply (simp add: executions_def reachable.reachable_0)
 
-  (* corresp_ex is execution, Lemma 2 *)
+  text \<open>\<open>corresp_ex\<close> is execution, Lemma 2\<close>
   apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
   apply (simp add: executions_def)
-  (* start state *)
+  text \<open>start state\<close>
   apply (rule conjI)
   apply (simp add: is_ref_map_def corresp_ex_def)
-  (* is-execution-fragment *)
+  text \<open>\<open>is_execution_fragment\<close>\<close>
   apply (erule lemma_2 [THEN spec, THEN mp])
   apply (simp add: reachable.reachable_0)
-
-done
+  done
 
 end
diff -r ec2f0dad8b98 -r 7355fd313cf8 src/HOL/HOLCF/IOA/ShortExecutions.thy
--- a/src/HOL/HOLCF/IOA/ShortExecutions.thy	Tue Jan 12 20:05:53 2016 +0100
+++ b/src/HOL/HOLCF/IOA/ShortExecutions.thy	Tue Jan 12 23:40:33 2016 +0100
@@ -13,39 +13,33 @@
   that has the same trace as ex, but its schedule ends with an external action.
 \<close>
 
-definition
-  oraclebuild :: "('a => bool) => 'a Seq -> 'a Seq -> 'a Seq" where
-  "oraclebuild P = (fix$(LAM h s t. case t of
-       nil => nil
-    | x##xs =>
-      (case x of
-        UU => UU
-      | Def y => (Takewhile (%x. \<not>P x)$s)
-                  @@ (y\<leadsto>(h$(TL$(Dropwhile (%x. \<not> P x)$s))$xs))
-      )
-    ))"
+definition oraclebuild :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Seq \<rightarrow> 'a Seq \<rightarrow> 'a Seq"
+  where "oraclebuild P =
+    (fix $
+      (LAM h s t.
+        case t of
+          nil \<Rightarrow> nil
+        | x ## xs \<Rightarrow>
+            (case x of
+              UU \<Rightarrow> UU
+            | Def y \<Rightarrow>
+                (Takewhile (\<lambda>x. \<not> P x) $ s) @@
+                (y \<leadsto> (h $ (TL $ (Dropwhile (\<lambda>x. \<not> P x) $ s)) $ xs)))))"
 
-definition
-  Cut :: "('a => bool) => 'a Seq => 'a Seq" where
-  "Cut P s = oraclebuild P$s$(Filter P$s)"
+definition Cut :: "('a \<Rightarrow> bool) \<Rightarrow> 'a Seq \<Rightarrow> 'a Seq"
+  where "Cut P s = oraclebuild P $ s $ (Filter P $ s)"
 
-definition
-  LastActExtsch :: "('a,'s)ioa => 'a Seq => bool" where
-  "LastActExtsch A sch = (Cut (%x. x: ext A) sch = sch)"
-
-(* LastActExtex      ::"('a,'s)ioa => ('a,'s) pairs  => bool"*)
-(* LastActExtex_def:
-  "LastActExtex A ex == LastActExtsch A (filter_act$ex)" *)
+definition LastActExtsch :: "('a,'s) ioa \<Rightarrow> 'a Seq \<Rightarrow> bool"
+  where "LastActExtsch A sch \<longleftrightarrow> Cut (\<lambda>x. x \<in> ext A) sch = sch"
 
 axiomatization where
-  Cut_prefixcl_Finite: "Finite s ==> (? y. s = Cut P s @@ y)"
+  Cut_prefixcl_Finite: "Finite s \<Longrightarrow> \<exists>y. s = Cut P s @@ y"
 
 axiomatization where
-  LastActExtsmall1: "LastActExtsch A sch ==> LastActExtsch A (TL$(Dropwhile P$sch))"
+  LastActExtsmall1: "LastActExtsch A sch \<Longrightarrow> LastActExtsch A (TL $ (Dropwhile P $ sch))"
 
 axiomatization where
-  LastActExtsmall2: "[| Finite sch1; LastActExtsch A (sch1 @@ sch2) |] ==> LastActExtsch A sch2"
-
+  LastActExtsmall2: "Finite sch1 \<Longrightarrow> LastActExtsch A (sch1 @@ sch2) \<Longrightarrow> LastActExtsch A sch2"
 
 ML \<open>
 fun thin_tac' ctxt j =
@@ -55,224 +49,209 @@
 \<close>
 
 
-subsection "oraclebuild rewrite rules"
-
+subsection \<open>\<open>oraclebuild\<close> rewrite rules\<close>
 
 lemma oraclebuild_unfold:
-"oraclebuild P = (LAM s t. case t of
-       nil => nil
-    | x##xs =>
-      (case x of
-        UU => UU
-      | Def y => (Takewhile (%a. \<not> P a)$s)
-                  @@ (y\<leadsto>(oraclebuild P$(TL$(Dropwhile (%a. \<not> P a)$s))$xs))
-      )
-    )"
-apply (rule trans)
-apply (rule fix_eq2)
-apply (simp only: oraclebuild_def)
-apply (rule beta_cfun)
-apply simp
-done
+  "oraclebuild P =
+    (LAM s t.
+      case t of
+        nil \<Rightarrow> nil
+      | x ## xs \<Rightarrow>
+          (case x of
+            UU \<Rightarrow> UU
+          | Def y \<Rightarrow>
+            (Takewhile (\<lambda>a. \<not> P a) $ s) @@
+            (y \<leadsto> (oraclebuild P $ (TL $ (Dropwhile (\<lambda>a. \<not> P a) $ s)) $ xs))))"
+  apply (rule trans)
+  apply (rule fix_eq2)
+  apply (simp only: oraclebuild_def)
+  apply (rule beta_cfun)
+  apply simp
+  done
 
-lemma oraclebuild_UU: "oraclebuild P$sch$UU = UU"
-apply (subst oraclebuild_unfold)
-apply simp
-done
+lemma oraclebuild_UU: "oraclebuild P $ sch $ UU = UU"
+  apply (subst oraclebuild_unfold)
+  apply simp
+  done
+
+lemma oraclebuild_nil: "oraclebuild P $ sch $ nil = nil"
+  apply (subst oraclebuild_unfold)
+  apply simp
+  done
 
-lemma oraclebuild_nil: "oraclebuild P$sch$nil = nil"
-apply (subst oraclebuild_unfold)
-apply simp
-done
-
-lemma oraclebuild_cons: "oraclebuild P$s$(x\<leadsto>t) =
-          (Takewhile (%a. \<not> P a)$s)
-           @@ (x\<leadsto>(oraclebuild P$(TL$(Dropwhile (%a. \<not> P a)$s))$t))"
-apply (rule trans)
-apply (subst oraclebuild_unfold)
-apply (simp add: Consq_def)
-apply (simp add: Consq_def)
-done
+lemma oraclebuild_cons:
+  "oraclebuild P $ s $ (x \<leadsto> t) =
+    (Takewhile (\<lambda>a. \<not> P a) $ s) @@
+    (x \<leadsto> (oraclebuild P $ (TL $ (Dropwhile (\<lambda>a. \<not> P a) $ s)) $ t))"
+  apply (rule trans)
+  apply (subst oraclebuild_unfold)
+  apply (simp add: Consq_def)
+  apply (simp add: Consq_def)
+  done
 
 
-subsection "Cut rewrite rules"
+subsection \<open>Cut rewrite rules\<close>
 
-lemma Cut_nil:
-"[| Forall (%a. \<not> P a) s; Finite s|]
-            ==> Cut P s =nil"
-apply (unfold Cut_def)
-apply (subgoal_tac "Filter P$s = nil")
-apply (simp (no_asm_simp) add: oraclebuild_nil)
-apply (rule ForallQFilterPnil)
-apply assumption+
-done
+lemma Cut_nil: "Forall (\<lambda>a. \<not> P a) s \<Longrightarrow> Finite s \<Longrightarrow> Cut P s = nil"
+  apply (unfold Cut_def)
+  apply (subgoal_tac "Filter P $ s = nil")
+  apply (simp add: oraclebuild_nil)
+  apply (rule ForallQFilterPnil)
+  apply assumption+
+  done
 
-lemma Cut_UU:
-"[| Forall (%a. \<not> P a) s; ~Finite s|]
-            ==> Cut P s =UU"
-apply (unfold Cut_def)
-apply (subgoal_tac "Filter P$s= UU")
-apply (simp (no_asm_simp) add: oraclebuild_UU)
-apply (rule ForallQFilterPUU)
-apply assumption+
-done
+lemma Cut_UU: "Forall (\<lambda>a. \<not> P a) s \<Longrightarrow> \<not> Finite s \<Longrightarrow> Cut P s = UU"
+  apply (unfold Cut_def)
+  apply (subgoal_tac "Filter P $ s = UU")
+  apply (simp add: oraclebuild_UU)
+  apply (rule ForallQFilterPUU)
+  apply assumption+
+  done
 
 lemma Cut_Cons:
-"[| P t;  Forall (%x. \<not> P x) ss; Finite ss|]
-            ==> Cut P (ss @@ (t\<leadsto> rs))
-                 = ss @@ (t \<leadsto> Cut P rs)"
-apply (unfold Cut_def)
-apply (simp add: ForallQFilterPnil oraclebuild_cons TakewhileConc DropwhileConc)
-done
+  "P t \<Longrightarrow> Forall (\<lambda>x. \<not> P x) ss \<Longrightarrow> Finite ss \<Longrightarrow>
+    Cut P (ss @@ (t \<leadsto> rs)) = ss @@ (t \<leadsto> Cut P rs)"
+  apply (unfold Cut_def)
+  apply (simp add: ForallQFilterPnil oraclebuild_cons TakewhileConc DropwhileConc)
+  done
 
 
-subsection "Cut lemmas for main theorem"
+subsection \<open>Cut lemmas for main theorem\<close>
 
-lemma FilterCut: "Filter P$s = Filter P$(Cut P s)"
-apply (rule_tac A1 = "%x. True" and Q1 = "%x. \<not> P x" and x1 = "s" in take_lemma_induct [THEN mp])
-prefer 3 apply (fast)
-apply (case_tac "Finite s")
-apply (simp add: Cut_nil ForallQFilterPnil)
-apply (simp add: Cut_UU ForallQFilterPUU)
-(* main case *)
-apply (simp add: Cut_Cons ForallQFilterPnil)
-done
-
+lemma FilterCut: "Filter P $ s = Filter P $ (Cut P s)"
+  apply (rule_tac A1 = "\<lambda>x. True" and Q1 = "\<lambda>x. \<not> P x" and x1 = "s" in take_lemma_induct [THEN mp])
+  prefer 3
+  apply fast
+  apply (case_tac "Finite s")
+  apply (simp add: Cut_nil ForallQFilterPnil)
+  apply (simp add: Cut_UU ForallQFilterPUU)
+  text \<open>main case\<close>
+  apply (simp add: Cut_Cons ForallQFilterPnil)
+  done
 
 lemma Cut_idemp: "Cut P (Cut P s) = (Cut P s)"
-apply (rule_tac A1 = "%x. True" and Q1 = "%x. \<not> P x" and x1 = "s" in
-  take_lemma_less_induct [THEN mp])
-prefer 3 apply (fast)
-apply (case_tac "Finite s")
-apply (simp add: Cut_nil ForallQFilterPnil)
-apply (simp add: Cut_UU ForallQFilterPUU)
-(* main case *)
-apply (simp add: Cut_Cons ForallQFilterPnil)
-apply (rule take_reduction)
-apply auto
-done
-
+  apply (rule_tac A1 = "\<lambda>x. True" and Q1 = "\<lambda>x. \<not> P x" and x1 = "s" in
+    take_lemma_less_induct [THEN mp])
+  prefer 3
+  apply fast
+  apply (case_tac "Finite s")
+  apply (simp add: Cut_nil ForallQFilterPnil)
+  apply (simp add: Cut_UU ForallQFilterPUU)
+  text \<open>main case\<close>
+  apply (simp add: Cut_Cons ForallQFilterPnil)
+  apply (rule take_reduction)
+  apply auto
+  done
 
 lemma MapCut: "Map f$(Cut (P \<circ> f) s) = Cut P (Map f$s)"
-apply (rule_tac A1 = "%x. True" and Q1 = "%x. \<not> P (f x) " and x1 = "s" in
-  take_lemma_less_induct [THEN mp])
-prefer 3 apply (fast)
-apply (case_tac "Finite s")
-apply (simp add: Cut_nil)
-apply (rule Cut_nil [symmetric])
-apply (simp add: ForallMap o_def)
-apply (simp add: Map2Finite)
-(* csae ~ Finite s *)
-apply (simp add: Cut_UU)
-apply (rule Cut_UU)
-apply (simp add: ForallMap o_def)
-apply (simp add: Map2Finite)
-(* main case *)
-apply (simp add: Cut_Cons MapConc ForallMap FiniteMap1 o_def)
-apply (rule take_reduction)
-apply auto
-done
-
+  apply (rule_tac A1 = "\<lambda>x. True" and Q1 = "\<lambda>x. \<not> P (f x) " and x1 = "s" in
+    take_lemma_less_induct [THEN mp])
+  prefer 3
+  apply fast
+  apply (case_tac "Finite s")
+  apply (simp add: Cut_nil)
+  apply (rule Cut_nil [symmetric])
+  apply (simp add: ForallMap o_def)
+  apply (simp add: Map2Finite)
+  text \<open>case \<open>\<not> Finite s\<close>\<close>
+  apply (simp add: Cut_UU)
+  apply (rule Cut_UU)
+  apply (simp add: ForallMap o_def)
+  apply (simp add: Map2Finite)
+  text \<open>main case\<close>
+  apply (simp add: Cut_Cons MapConc ForallMap FiniteMap1 o_def)
+  apply (rule take_reduction)
+  apply auto
+  done
 
-lemma Cut_prefixcl_nFinite [rule_format (no_asm)]: "~Finite s --> Cut P s << s"
-apply (intro strip)
-apply (rule take_lemma_less [THEN iffD1])
-apply (intro strip)
-apply (rule mp)
-prefer 2 apply (assumption)
-apply (tactic "thin_tac' @{context} 1 1")
-apply (rule_tac x = "s" in spec)
-apply (rule nat_less_induct)
-apply (intro strip)
-apply (rename_tac na n s)
-apply (case_tac "Forall (%x. ~ P x) s")
-apply (rule take_lemma_less [THEN iffD2, THEN spec])
-apply (simp add: Cut_UU)
-(* main case *)
-apply (drule divide_Seq3)
-apply (erule exE)+
-apply (erule conjE)+
-apply hypsubst
-apply (simp add: Cut_Cons)
-apply (rule take_reduction_less)
-(* auto makes also reasoning about Finiteness of parts of s ! *)
-apply auto
-done
+lemma Cut_prefixcl_nFinite [rule_format]: "\<not> Finite s \<longrightarrow> Cut P s \<sqsubseteq> s"
+  apply (intro strip)
+  apply (rule take_lemma_less [THEN iffD1])
+  apply (intro strip)
+  apply (rule mp)
+  prefer 2
+  apply assumption
+  apply (tactic "thin_tac' @{context} 1 1")
+  apply (rule_tac x = "s" in spec)
+  apply (rule nat_less_induct)
+  apply (intro strip)
+  apply (rename_tac na n s)
+  apply (case_tac "Forall (\<lambda>x. \<not> P x) s")
+  apply (rule take_lemma_less [THEN iffD2, THEN spec])
+  apply (simp add: Cut_UU)
+  text \<open>main case\<close>
+  apply (drule divide_Seq3)
+  apply (erule exE)+
+  apply (erule conjE)+
+  apply hypsubst
+  apply (simp add: Cut_Cons)
+  apply (rule take_reduction_less)
+  text \<open>auto makes also reasoning about Finiteness of parts of \<open>s\<close>!\<close>
+  apply auto
+  done
 
-
-lemma execThruCut: "!!ex .is_exec_frag A (s,ex) ==> is_exec_frag A (s,Cut P ex)"
-apply (case_tac "Finite ex")
-apply (cut_tac s = "ex" and P = "P" in Cut_prefixcl_Finite)
-apply assumption
-apply (erule exE)
-apply (rule exec_prefix2closed)
-apply (erule_tac s = "ex" and t = "Cut P ex @@ y" in subst)
-apply assumption
-apply (erule exec_prefixclosed)
-apply (erule Cut_prefixcl_nFinite)
-done
+lemma execThruCut: "is_exec_frag A (s, ex) \<Longrightarrow> is_exec_frag A (s, Cut P ex)"
+  apply (case_tac "Finite ex")
+  apply (cut_tac s = "ex" and P = "P" in Cut_prefixcl_Finite)
+  apply assumption
+  apply (erule exE)
+  apply (rule exec_prefix2closed)
+  apply (erule_tac s = "ex" and t = "Cut P ex @@ y" in subst)
+  apply assumption
+  apply (erule exec_prefixclosed)
+  apply (erule Cut_prefixcl_nFinite)
+  done
 
 
-subsection "Main Cut Theorem"
+subsection \<open>Main Cut Theorem\<close>
 
 lemma exists_LastActExtsch:
- "[|sch : schedules A ; tr = Filter (%a. a:ext A)$sch|]
-    ==> ? sch. sch : schedules A &
-               tr = Filter (%a. a:ext A)$sch &
-               LastActExtsch A sch"
-
-apply (unfold schedules_def has_schedule_def [abs_def])
-apply auto
-apply (rule_tac x = "filter_act$ (Cut (%a. fst a:ext A) (snd ex))" in exI)
-apply (simp add: executions_def)
-apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
-apply auto
-apply (rule_tac x = " (x1,Cut (%a. fst a:ext A) x2) " in exI)
-apply (simp (no_asm_simp))
-
-(* Subgoal 1: Lemma:  propagation of execution through Cut *)
+  "sch \<in> schedules A \<Longrightarrow> tr = Filter (\<lambda>a. a \<in> ext A) $ sch \<Longrightarrow>
+    \<exists>sch. sch \<in> schedules A \<and> tr = Filter (\<lambda>a. a \<in> ext A) $ sch \<and> LastActExtsch A sch"
+  apply (unfold schedules_def has_schedule_def [abs_def])
+  apply auto
+  apply (rule_tac x = "filter_act $ (Cut (\<lambda>a. fst a \<in> ext A) (snd ex))" in exI)
+  apply (simp add: executions_def)
+  apply (tactic \<open>pair_tac @{context} "ex" 1\<close>)
+  apply auto
+  apply (rule_tac x = "(x1, Cut (\<lambda>a. fst a \<in> ext A) x2)" in exI)
+  apply simp
 
-apply (simp add: execThruCut)
-
-(* Subgoal 2:  Lemma:  Filter P s = Filter P (Cut P s) *)
-
-apply (simp (no_asm) add: filter_act_def)
-apply (subgoal_tac "Map fst$ (Cut (%a. fst a: ext A) x2) = Cut (%a. a:ext A) (Map fst$x2) ")
+  text \<open>Subgoal 1: Lemma:  propagation of execution through \<open>Cut\<close>\<close>
+  apply (simp add: execThruCut)
 
-apply (rule_tac [2] MapCut [unfolded o_def])
-apply (simp add: FilterCut [symmetric])
-
-(* Subgoal 3: Lemma: Cut idempotent  *)
+  text \<open>Subgoal 2:  Lemma:  \<open>Filter P s = Filter P (Cut P s)\<close>\<close>
+  apply (simp add: filter_act_def)
+  apply (subgoal_tac "Map fst $ (Cut (\<lambda>a. fst a \<in> ext A) x2) = Cut (\<lambda>a. a \<in> ext A) (Map fst $ x2)")
+  apply (rule_tac [2] MapCut [unfolded o_def])
+  apply (simp add: FilterCut [symmetric])
 
-apply (simp (no_asm) add: LastActExtsch_def filter_act_def)
-apply (subgoal_tac "Map fst$ (Cut (%a. fst a: ext A) x2) = Cut (%a. a:ext A) (Map fst$x2) ")
-apply (rule_tac [2] MapCut [unfolded o_def])
-apply (simp add: Cut_idemp)
-done
+  text \<open>Subgoal 3: Lemma: \<open>Cut\<close> idempotent\<close>
+  apply (simp add: LastActExtsch_def filter_act_def)
+  apply (subgoal_tac "Map fst $ (Cut (\<lambda>a. fst a \<in> ext A) x2) = Cut (\<lambda>a. a \<in> ext A) (Map fst $ x2)")
+  apply (rule_tac [2] MapCut [unfolded o_def])
+  apply (simp add: Cut_idemp)
+  done
 
 
-subsection "Further Cut lemmas"
+subsection \<open>Further Cut lemmas\<close>
 
-lemma LastActExtimplnil:
-  "[| LastActExtsch A sch; Filter (%x. x:ext A)$sch = nil |]
-    ==> sch=nil"
-apply (unfold LastActExtsch_def)
-apply (drule FilternPnilForallP)
-apply (erule conjE)
-apply (drule Cut_nil)
-apply assumption
-apply simp
-done
+lemma LastActExtimplnil: "LastActExtsch A sch \<Longrightarrow> Filter (\<lambda>x. x \<in> ext A) $ sch = nil \<Longrightarrow> sch = nil"
+  apply (unfold LastActExtsch_def)
+  apply (drule FilternPnilForallP)
+  apply (erule conjE)
+  apply (drule Cut_nil)
+  apply assumption
+  apply simp
+  done
 
-lemma LastActExtimplUU:
-  "[| LastActExtsch A sch; Filter (%x. x:ext A)$sch = UU |]
-    ==> sch=UU"
-apply (unfold LastActExtsch_def)
-apply (drule FilternPUUForallP)
-apply (erule conjE)
-apply (drule Cut_UU)
-apply assumption
-apply simp
-done
+lemma LastActExtimplUU: "LastActExtsch A sch \<Longrightarrow> Filter (\<lambda>x. x \<in> ext A) $ sch = UU \<Longrightarrow> sch = UU"
+  apply (unfold LastActExtsch_def)
+  apply (drule FilternPUUForallP)
+  apply (erule conjE)
+  apply (drule Cut_UU)
+  apply assumption
+  apply simp
+  done
 
 end