src/HOL/HOLCF/IOA/meta_theory/Automata.thy

 imports Asig
 begin
 
 default_sort type
 
-type_synonym
-  ('a, 's) transition = "'s * 'a * 's"
-
-type_synonym
-  ('a, 's) ioa = "'a signature * 's set * ('a,'s)transition set * ('a set set) * ('a set set)"
-
-consts
-
-  (* IO automata *)
-
-  asig_of        ::"('a,'s)ioa => 'a signature"
-  starts_of      ::"('a,'s)ioa => 's set"
-  trans_of       ::"('a,'s)ioa => ('a,'s)transition set"
-  wfair_of       ::"('a,'s)ioa => ('a set) set"
-  sfair_of       ::"('a,'s)ioa => ('a set) set"
-
-  is_asig_of     ::"('a,'s)ioa => bool"
-  is_starts_of   ::"('a,'s)ioa => bool"
-  is_trans_of    ::"('a,'s)ioa => bool"
-  input_enabled  ::"('a,'s)ioa => bool"
-  IOA            ::"('a,'s)ioa => bool"
-
-  (* constraints for fair IOA *)
-
-  fairIOA        ::"('a,'s)ioa => bool"
-  input_resistant::"('a,'s)ioa => bool"
-
-  (* enabledness of actions and action sets *)
-
-  enabled        ::"('a,'s)ioa => 'a => 's => bool"
-  Enabled    ::"('a,'s)ioa => 'a set => 's => bool"
-
-  (* action set keeps enabled until probably disabled by itself *)
-
-  en_persistent  :: "('a,'s)ioa => 'a set => bool"
-
- (* post_conditions for actions and action sets *)
-
-  was_enabled        ::"('a,'s)ioa => 'a => 's => bool"
-  set_was_enabled    ::"('a,'s)ioa => 'a set => 's => bool"
-
-  (* invariants *)
-  invariant     :: "[('a,'s)ioa, 's=>bool] => bool"
-
-  (* binary composition of action signatures and automata *)
-  asig_comp    ::"['a signature, 'a signature] => 'a signature"
-  compatible   ::"[('a,'s)ioa, ('a,'t)ioa] => bool"
-  par          ::"[('a,'s)ioa, ('a,'t)ioa] => ('a,'s*'t)ioa"  (infixr "\<parallel>" 10)
-
-  (* hiding and restricting *)
-  hide_asig     :: "['a signature, 'a set] => 'a signature"
-  hide          :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
-  restrict_asig :: "['a signature, 'a set] => 'a signature"
-  restrict      :: "[('a,'s)ioa, 'a set] => ('a,'s)ioa"
-
-  (* renaming *)
-  rename_set    :: "'a set => ('c => 'a option) => 'c set"
-  rename        :: "('a, 'b)ioa => ('c => 'a option) => ('c,'b)ioa"
-
-
-inductive
-  reachable :: "('a, 's) ioa => 's => bool"
-  for C :: "('a, 's) ioa"
-  where
-    reachable_0:  "s : starts_of C ==> reachable C s"
-  | reachable_n:  "[| reachable C s; (s, a, t) : trans_of C |] ==> reachable C t"
-
-abbreviation
-  trans_of_syn  ("_ \<midarrow>_\<midarrow>_\<rightarrow> _" [81,81,81,81] 100) where
-  "s \<midarrow>a\<midarrow>A\<rightarrow> t == (s,a,t):trans_of A"
+type_synonym ('a, 's) transition = "'s * 'a * 's"
+type_synonym ('a, 's) ioa =
+  "'a signature * 's set * ('a,'s)transition set * ('a set set) * ('a set set)"
+
+
+(* --------------------------------- IOA ---------------------------------*)
+
+(* IO automata *)
+
+definition asig_of :: "('a, 's)ioa \<Rightarrow> 'a signature"
+  where "asig_of = fst"
+
+definition starts_of :: "('a, 's) ioa \<Rightarrow> 's set"
+  where "starts_of = (fst \<circ> snd)"
+
+definition trans_of :: "('a, 's) ioa \<Rightarrow> ('a, 's) transition set"
+  where "trans_of = (fst \<circ> snd \<circ> snd)"
+
+abbreviation trans_of_syn  ("_ \<midarrow>_\<midarrow>_\<rightarrow> _" [81, 81, 81, 81] 100)
+  where "s \<midarrow>a\<midarrow>A\<rightarrow> t \<equiv> (s, a, t) \<in> trans_of A"
+
+definition wfair_of :: "('a, 's) ioa \<Rightarrow> 'a set set"
+  where "wfair_of = (fst \<circ> snd \<circ> snd \<circ> snd)"
+
+definition sfair_of :: "('a, 's) ioa \<Rightarrow> 'a set set"
+  where "sfair_of = (snd \<circ> snd \<circ> snd \<circ> snd)"
+
+definition is_asig_of :: "('a, 's) ioa \<Rightarrow> bool"
+  where "is_asig_of A = is_asig (asig_of A)"
+
+definition is_starts_of :: "('a, 's) ioa \<Rightarrow> bool"
+  where "is_starts_of A \<longleftrightarrow> starts_of A \<noteq> {}"
+
+definition is_trans_of :: "('a, 's) ioa \<Rightarrow> bool"
+  where "is_trans_of A \<longleftrightarrow>
+    (\<forall>triple. triple \<in> trans_of A \<longrightarrow> fst (snd triple) \<in> actions (asig_of A))"
+
+definition input_enabled :: "('a, 's) ioa \<Rightarrow> bool"
+  where "input_enabled A \<longleftrightarrow>
+    (\<forall>a. a \<in> inputs (asig_of A) \<longrightarrow> (\<forall>s1. \<exists>s2. (s1, a, s2) \<in> trans_of A))"
+
+definition IOA :: "('a, 's) ioa \<Rightarrow> bool"
+  where "IOA A \<longleftrightarrow>
+    is_asig_of A \<and>
+    is_starts_of A \<and>
+    is_trans_of A \<and>
+    input_enabled A"
 
 abbreviation "act A == actions (asig_of A)"
 abbreviation "ext A == externals (asig_of A)"
 abbreviation int where "int A == internals (asig_of A)"
 abbreviation "inp A == inputs (asig_of A)"
 abbreviation "out A == outputs (asig_of A)"
 abbreviation "local A == locals (asig_of A)"
 
-defs
-
-(* --------------------------------- IOA ---------------------------------*)
-
-asig_of_def:   "asig_of == fst"
-starts_of_def: "starts_of == (fst o snd)"
-trans_of_def:  "trans_of == (fst o snd o snd)"
-wfair_of_def:  "wfair_of == (fst o snd o snd o snd)"
-sfair_of_def:  "sfair_of == (snd o snd o snd o snd)"
-
-is_asig_of_def:
-  "is_asig_of A == is_asig (asig_of A)"
-
-is_starts_of_def:
-  "is_starts_of A ==  (~ starts_of A = {})"
-
-is_trans_of_def:
-  "is_trans_of A ==
-    (!triple. triple:(trans_of A) --> fst(snd(triple)):actions(asig_of A))"
-
-input_enabled_def:
-  "input_enabled A ==
-    (!a. (a:inputs(asig_of A)) --> (!s1. ? s2. (s1,a,s2):(trans_of A)))"
-
-
-ioa_def:
-  "IOA A == (is_asig_of A    &
-             is_starts_of A  &
-             is_trans_of A   &
-             input_enabled A)"
-
-
-invariant_def: "invariant A P == (!s. reachable A s --> P(s))"
+(* invariants *)
+inductive reachable :: "('a, 's) ioa \<Rightarrow> 's \<Rightarrow> bool"
+  for C :: "('a, 's) ioa"
+where
+  reachable_0:  "s \<in> starts_of C \<Longrightarrow> reachable C s"
+| reachable_n:  "\<lbrakk>reachable C s; (s, a, t) \<in> trans_of C\<rbrakk> \<Longrightarrow> reachable C t"
+
+definition invariant :: "[('a, 's) ioa, 's \<Rightarrow> bool] \<Rightarrow> bool"
+  where "invariant A P \<longleftrightarrow> (\<forall>s. reachable A s \<longrightarrow> P s)"
 
 
 (* ------------------------- parallel composition --------------------------*)
 
-
-compatible_def:
-  "compatible A B ==
-  (((out A Int out B) = {}) &
-   ((int A Int act B) = {}) &
-   ((int B Int act A) = {}))"
-
-asig_comp_def:
-  "asig_comp a1 a2 ==
-     (((inputs(a1) Un inputs(a2)) - (outputs(a1) Un outputs(a2)),
-       (outputs(a1) Un outputs(a2)),
-       (internals(a1) Un internals(a2))))"
-
-par_def:
-  "(A \<parallel> B) ==
+(* binary composition of action signatures and automata *)
+
+definition compatible :: "[('a, 's) ioa, ('a, 't) ioa] \<Rightarrow> bool"
+where
+  "compatible A B \<longleftrightarrow>
+  (((out A \<inter> out B) = {}) \<and>
+   ((int A \<inter> act B) = {}) \<and>
+   ((int B \<inter> act A) = {}))"
+
+definition asig_comp :: "['a signature, 'a signature] \<Rightarrow> 'a signature"
+where
+  "asig_comp a1 a2 =
+     (((inputs(a1) \<union> inputs(a2)) - (outputs(a1) \<union> outputs(a2)),
+       (outputs(a1) \<union> outputs(a2)),
+       (internals(a1) \<union> internals(a2))))"
+
+definition par :: "[('a, 's) ioa, ('a, 't) ioa] \<Rightarrow> ('a, 's * 't) ioa"  (infixr "\<parallel>" 10)
+where
+  "(A \<parallel> B) =
       (asig_comp (asig_of A) (asig_of B),
-       {pr. fst(pr):starts_of(A) & snd(pr):starts_of(B)},
+       {pr. fst(pr) \<in> starts_of(A) \<and> snd(pr) \<in> starts_of(B)},
        {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))
-            in (a:act A | a:act B) &
-               (if a:act A then
-                  (fst(s),a,fst(t)):trans_of(A)
+            in (a \<in> act A | a:act B) \<and>
+               (if a \<in> act A then
+                  (fst(s), a, fst(t)) \<in> trans_of(A)
                 else fst(t) = fst(s))
                &
-               (if a:act B then
-                  (snd(s),a,snd(t)):trans_of(B)
+               (if a \<in> act B then
+                  (snd(s), a, snd(t)) \<in> trans_of(B)
                 else snd(t) = snd(s))},
-        wfair_of A Un wfair_of B,
-        sfair_of A Un sfair_of B)"
+        wfair_of A \<union> wfair_of B,
+        sfair_of A \<union> sfair_of B)"
 
 
 (* ------------------------ hiding -------------------------------------------- *)
 
-restrict_asig_def:
-  "restrict_asig asig actns ==
+(* hiding and restricting *)
+
+definition restrict_asig :: "['a signature, 'a set] \<Rightarrow> 'a signature"
+where
+  "restrict_asig asig actns =
     (inputs(asig) Int actns,
      outputs(asig) Int actns,
      internals(asig) Un (externals(asig) - actns))"
 
 (* Notice that for wfair_of and sfair_of nothing has to be changed, as
    changes from the outputs to the internals does not touch the locals as
    a whole, which is of importance for fairness only *)
-
-restrict_def:
-  "restrict A actns ==
+definition restrict :: "[('a, 's) ioa, 'a set] \<Rightarrow> ('a, 's) ioa"
+where
+  "restrict A actns =
     (restrict_asig (asig_of A) actns,
      starts_of A,
      trans_of A,
      wfair_of A,
      sfair_of A)"
 
-hide_asig_def:
-  "hide_asig asig actns ==
+definition hide_asig :: "['a signature, 'a set] \<Rightarrow> 'a signature"
+where
+  "hide_asig asig actns =
     (inputs(asig) - actns,
      outputs(asig) - actns,
-     internals(asig) Un actns)"
-
-hide_def:
-  "hide A actns ==
+     internals(asig) \<union> actns)"
+
+definition hide :: "[('a, 's) ioa, 'a set] \<Rightarrow> ('a, 's) ioa"
+where
+  "hide A actns =
     (hide_asig (asig_of A) actns,
      starts_of A,
      trans_of A,
      wfair_of A,
      sfair_of A)"
 
 (* ------------------------- renaming ------------------------------------------- *)
 
-rename_set_def:
-  "rename_set A ren == {b. ? x. Some x = ren b & x : A}"
-
-rename_def:
-"rename ioa ren ==
-  ((rename_set (inp ioa) ren,
-    rename_set (out ioa) ren,
-    rename_set (int ioa) ren),
-   starts_of ioa,
-   {tr. let s = fst(tr); a = fst(snd(tr));  t = snd(snd(tr))
-        in
-        ? x. Some(x) = ren(a) & (s,x,t):trans_of ioa},
-   {rename_set s ren | s. s: wfair_of ioa},
-   {rename_set s ren | s. s: sfair_of ioa})"
+definition rename_set :: "'a set \<Rightarrow> ('c \<Rightarrow> 'a option) \<Rightarrow> 'c set"
+  where "rename_set A ren = {b. \<exists>x. Some x = ren b \<and> x \<in> A}"
+
+definition rename :: "('a, 'b) ioa \<Rightarrow> ('c \<Rightarrow> 'a option) \<Rightarrow> ('c, 'b) ioa"
+where
+  "rename ioa ren =
+    ((rename_set (inp ioa) ren,
+      rename_set (out ioa) ren,
+      rename_set (int ioa) ren),
+     starts_of ioa,
+     {tr. let s = fst(tr); a = fst(snd(tr));  t = snd(snd(tr))
+          in
+          \<exists>x. Some(x) = ren(a) \<and> (s,x,t):trans_of ioa},
+     {rename_set s ren | s. s \<in> wfair_of ioa},
+     {rename_set s ren | s. s \<in> sfair_of ioa})"
+
 
 (* ------------------------- fairness ----------------------------- *)
 
-fairIOA_def:
-  "fairIOA A == (! S : wfair_of A. S<= local A) &
-                (! S : sfair_of A. S<= local A)"
-
-input_resistant_def:
-  "input_resistant A == ! W : sfair_of A. ! s a t.
-                        reachable A s & reachable A t & a:inp A &
-                        Enabled A W s & s \<midarrow>a\<midarrow>A\<rightarrow> t
-                        --> Enabled A W t"
-
-enabled_def:
-  "enabled A a s == ? t. s \<midarrow>a\<midarrow>A\<rightarrow> t"
-
-Enabled_def:
-  "Enabled A W s == ? w:W. enabled A w s"
-
-en_persistent_def:
-  "en_persistent A W == ! s a t. Enabled A W s &
-                                 a ~:W &
-                                 s \<midarrow>a\<midarrow>A\<rightarrow> t
-                                 --> Enabled A W t"
-was_enabled_def:
-  "was_enabled A a t == ? s. s \<midarrow>a\<midarrow>A\<rightarrow> t"
-
-set_was_enabled_def:
-  "set_was_enabled A W t == ? w:W. was_enabled A w t"
+(* enabledness of actions and action sets *)
+
+definition enabled :: "('a, 's) ioa \<Rightarrow> 'a \<Rightarrow> 's \<Rightarrow> bool"
+  where "enabled A a s \<longleftrightarrow> (\<exists>t. s \<midarrow>a\<midarrow>A\<rightarrow> t)"
+
+definition Enabled :: "('a, 's) ioa \<Rightarrow> 'a set \<Rightarrow> 's \<Rightarrow> bool"
+  where "Enabled A W s \<longleftrightarrow> (\<exists>w \<in> W. enabled A w s)"
+
+
+(* action set keeps enabled until probably disabled by itself *)
+
+definition en_persistent :: "('a, 's) ioa \<Rightarrow> 'a set \<Rightarrow> bool"
+where
+  "en_persistent A W \<longleftrightarrow>
+    (\<forall>s a t. Enabled A W s \<and> a \<notin> W \<and> s \<midarrow>a\<midarrow>A\<rightarrow> t \<longrightarrow> Enabled A W t)"
+
+
+(* post_conditions for actions and action sets *)
+
+definition was_enabled :: "('a, 's) ioa \<Rightarrow> 'a \<Rightarrow> 's \<Rightarrow> bool"
+  where "was_enabled A a t \<longleftrightarrow> (\<exists>s. s \<midarrow>a\<midarrow>A\<rightarrow> t)"
+
+definition set_was_enabled :: "('a, 's) ioa \<Rightarrow> 'a set \<Rightarrow> 's \<Rightarrow> bool"
+  where "set_was_enabled A W t \<longleftrightarrow> (\<exists>w \<in> W. was_enabled A w t)"
+
+
+(* constraints for fair IOA *)
+
+definition fairIOA :: "('a, 's) ioa \<Rightarrow> bool"
+  where "fairIOA A \<longleftrightarrow> (\<forall>S \<in> wfair_of A. S \<subseteq> local A) \<and> (\<forall>S \<in> sfair_of A. S \<subseteq> local A)"
+
+definition input_resistant :: "('a, 's) ioa \<Rightarrow> bool"
+where
+  "input_resistant A \<longleftrightarrow>
+    (\<forall>W \<in> sfair_of A. \<forall>s a t.
+      reachable A s \<and> reachable A t \<and> a \<in> inp A \<and>
+      Enabled A W s \<and> s \<midarrow>a\<midarrow>A\<rightarrow> t \<longrightarrow> Enabled A W t)"
 
 
 declare split_paired_Ex [simp del]
 
 lemmas ioa_projections = asig_of_def starts_of_def trans_of_def wfair_of_def sfair_of_def
 
 
 subsection "asig_of, starts_of, trans_of"
 
-lemma ioa_triple_proj: 
- "((asig_of (x,y,z,w,s)) = x)   &  
-  ((starts_of (x,y,z,w,s)) = y) &  
-  ((trans_of (x,y,z,w,s)) = z)  &  
-  ((wfair_of (x,y,z,w,s)) = w) &  
+lemma ioa_triple_proj:
+ "((asig_of (x,y,z,w,s)) = x)   &
+  ((starts_of (x,y,z,w,s)) = y) &
+  ((trans_of (x,y,z,w,s)) = z)  &
+  ((wfair_of (x,y,z,w,s)) = w) &
   ((sfair_of (x,y,z,w,s)) = s)"
   apply (simp add: ioa_projections)
   done
 
-lemma trans_in_actions: 
+lemma trans_in_actions:
   "[| is_trans_of A; (s1,a,s2):trans_of(A) |] ==> a:act A"
-apply (unfold is_trans_of_def actions_def is_asig_def)
-  apply (erule allE, erule impE, assumption)
-  apply simp
-done
-
-lemma starts_of_par: 
-"starts_of(A \<parallel> B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}"
-  apply (simp add: par_def ioa_projections)
-done
-
-lemma trans_of_par: 
-"trans_of(A \<parallel> B) = {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))  
-             in (a:act A | a:act B) &  
-                (if a:act A then        
-                   (fst(s),a,fst(t)):trans_of(A)  
-                 else fst(t) = fst(s))             
-                &                                   
-                (if a:act B then                     
-                   (snd(s),a,snd(t)):trans_of(B)      
+  apply (unfold is_trans_of_def actions_def is_asig_def)
+    apply (erule allE, erule impE, assumption)
+    apply simp
+  done
+
+lemma starts_of_par: "starts_of(A \<parallel> B) = {p. fst(p):starts_of(A) & snd(p):starts_of(B)}"
+  by (simp add: par_def ioa_projections)
+
+lemma trans_of_par:
+"trans_of(A \<parallel> B) = {tr. let s = fst(tr); a = fst(snd(tr)); t = snd(snd(tr))
+             in (a:act A | a:act B) &
+                (if a:act A then
+                   (fst(s),a,fst(t)):trans_of(A)
+                 else fst(t) = fst(s))
+                &
+                (if a:act B then
+                   (snd(s),a,snd(t)):trans_of(B)
                  else snd(t) = snd(s))}"
-
-apply (simp add: par_def ioa_projections)
-done
+  by (simp add: par_def ioa_projections)
 
 
 subsection "actions and par"
 
-lemma actions_asig_comp: 
-  "actions(asig_comp a b) = actions(a) Un actions(b)"
-  apply (simp (no_asm) add: actions_def asig_comp_def asig_projections)
-  apply blast
-  done
+lemma actions_asig_comp: "actions(asig_comp a b) = actions(a) Un actions(b)"
+  by (auto simp add: actions_def asig_comp_def asig_projections)
 
 lemma asig_of_par: "asig_of(A \<parallel> B) = asig_comp (asig_of A) (asig_of B)"
-  apply (simp add: par_def ioa_projections)
-  done
-
-
-lemma externals_of_par: "ext (A1\<parallel>A2) =     
-   (ext A1) Un (ext A2)"
-apply (simp add: externals_def asig_of_par asig_comp_def
-  asig_inputs_def asig_outputs_def Un_def set_diff_eq)
-apply blast
-done
-
-lemma actions_of_par: "act (A1\<parallel>A2) =     
-   (act A1) Un (act A2)"
-apply (simp add: actions_def asig_of_par asig_comp_def
-  asig_inputs_def asig_outputs_def asig_internals_def Un_def set_diff_eq)
-apply blast
-done
-
-lemma inputs_of_par: "inp (A1\<parallel>A2) = 
-          ((inp A1) Un (inp A2)) - ((out A1) Un (out A2))"
-apply (simp add: actions_def asig_of_par asig_comp_def
-  asig_inputs_def asig_outputs_def Un_def set_diff_eq)
-done
-
-lemma outputs_of_par: "out (A1\<parallel>A2) = 
-          (out A1) Un (out A2)"
-apply (simp add: actions_def asig_of_par asig_comp_def
-  asig_outputs_def Un_def set_diff_eq)
-done
-
-lemma internals_of_par: "int (A1\<parallel>A2) = 
-          (int A1) Un (int A2)"
-apply (simp add: actions_def asig_of_par asig_comp_def
-  asig_inputs_def asig_outputs_def asig_internals_def Un_def set_diff_eq)
-done
+  by (simp add: par_def ioa_projections)
+
+
+lemma externals_of_par: "ext (A1\<parallel>A2) = (ext A1) Un (ext A2)"
+  apply (simp add: externals_def asig_of_par asig_comp_def
+    asig_inputs_def asig_outputs_def Un_def set_diff_eq)
+  apply blast
+  done
+
+lemma actions_of_par: "act (A1\<parallel>A2) = (act A1) Un (act A2)"
+  apply (simp add: actions_def asig_of_par asig_comp_def
+    asig_inputs_def asig_outputs_def asig_internals_def Un_def set_diff_eq)
+  apply blast
+  done
+
+lemma inputs_of_par: "inp (A1\<parallel>A2) = ((inp A1) Un (inp A2)) - ((out A1) Un (out A2))"
+  by (simp add: actions_def asig_of_par asig_comp_def
+    asig_inputs_def asig_outputs_def Un_def set_diff_eq)
+
+lemma outputs_of_par: "out (A1\<parallel>A2) = (out A1) Un (out A2)"
+  by (simp add: actions_def asig_of_par asig_comp_def
+    asig_outputs_def Un_def set_diff_eq)
+
+lemma internals_of_par: "int (A1\<parallel>A2) = (int A1) Un (int A2)"
+  by (simp add: actions_def asig_of_par asig_comp_def
+    asig_inputs_def asig_outputs_def asig_internals_def Un_def set_diff_eq)
 
 
 subsection "actions and compatibility"
 
 lemma compat_commute: "compatible A B = compatible B A"
-apply (simp add: compatible_def Int_commute)
-apply auto
-done
-
-lemma ext1_is_not_int2: 
- "[| compatible A1 A2; a:ext A1|] ==> a~:int A2"
-apply (unfold externals_def actions_def compatible_def)
-apply simp
-apply blast
-done
+  by (auto simp add: compatible_def Int_commute)
+
+lemma ext1_is_not_int2: "[| compatible A1 A2; a:ext A1|] ==> a~:int A2"
+  apply (unfold externals_def actions_def compatible_def)
+  apply simp
+  apply blast
+  done
 
 (* just commuting the previous one: better commute compatible *)
-lemma ext2_is_not_int1: 
- "[| compatible A2 A1 ; a:ext A1|] ==> a~:int A2"
-apply (unfold externals_def actions_def compatible_def)
-apply simp
-apply blast
-done
+lemma ext2_is_not_int1: "[| compatible A2 A1 ; a:ext A1|] ==> a~:int A2"
+  apply (unfold externals_def actions_def compatible_def)
+  apply simp
+  apply blast
+  done
 
 lemmas ext1_ext2_is_not_act2 = ext1_is_not_int2 [THEN int_and_ext_is_act]
 lemmas ext1_ext2_is_not_act1 = ext2_is_not_int1 [THEN int_and_ext_is_act]
 
-lemma intA_is_not_extB: 
- "[| compatible A B; x:int A |] ==> x~:ext B"
-apply (unfold externals_def actions_def compatible_def)
-apply simp
-apply blast
-done
-
-lemma intA_is_not_actB: 
-"[| compatible A B; a:int A |] ==> a ~: act B"
-apply (unfold externals_def actions_def compatible_def is_asig_def asig_of_def)
-apply simp
-apply blast
-done
+lemma intA_is_not_extB: "[| compatible A B; x:int A |] ==> x~:ext B"
+  apply (unfold externals_def actions_def compatible_def)
+  apply simp
+  apply blast
+  done
+
+lemma intA_is_not_actB: "[| compatible A B; a:int A |] ==> a ~: act B"
+  apply (unfold externals_def actions_def compatible_def is_asig_def asig_of_def)
+  apply simp
+  apply blast
+  done
 
 (* the only one that needs disjointness of outputs and of internals and _all_ acts *)
-lemma outAactB_is_inpB: 
-"[| compatible A B; a:out A ;a:act B|] ==> a : inp B"
-apply (unfold asig_outputs_def asig_internals_def actions_def asig_inputs_def 
-    compatible_def is_asig_def asig_of_def)
-apply simp
-apply blast
-done
+lemma outAactB_is_inpB: "[| compatible A B; a:out A ;a:act B|] ==> a : inp B"
+  apply (unfold asig_outputs_def asig_internals_def actions_def asig_inputs_def
+      compatible_def is_asig_def asig_of_def)
+  apply simp
+  apply blast
+  done
 
 (* needed for propagation of input_enabledness from A,B to A\<parallel>B *)
-lemma inpAAactB_is_inpBoroutB: 
-"[| compatible A B; a:inp A ;a:act B|] ==> a : inp B | a: out B"
-apply (unfold asig_outputs_def asig_internals_def actions_def asig_inputs_def 
-    compatible_def is_asig_def asig_of_def)
-apply simp
-apply blast
-done
+lemma inpAAactB_is_inpBoroutB:
+  "[| compatible A B; a:inp A ;a:act B|] ==> a : inp B | a: out B"
+  apply (unfold asig_outputs_def asig_internals_def actions_def asig_inputs_def
+      compatible_def is_asig_def asig_of_def)
+  apply simp
+  apply blast
+  done
 
 
 subsection "input_enabledness and par"
 
 (* ugly case distinctions. Heart of proof:
      1. inpAAactB_is_inpBoroutB ie. internals are really hidden.
      2. inputs_of_par: outputs are no longer inputs of par. This is important here *)
-lemma input_enabled_par: 
-"[| compatible A B; input_enabled A; input_enabled B|]  
-      ==> input_enabled (A\<parallel>B)"
-apply (unfold input_enabled_def)
-apply (simp add: Let_def inputs_of_par trans_of_par)
-apply (tactic "safe_tac (Context.raw_transfer @{theory} @{theory_context Fun})")
-apply (simp add: inp_is_act)
-prefer 2
-apply (simp add: inp_is_act)
-(* a: inp A *)
-apply (case_tac "a:act B")
-(* a:act B *)
-apply (erule_tac x = "a" in allE)
-apply simp
-apply (drule inpAAactB_is_inpBoroutB)
-apply assumption
-apply assumption
-apply (erule_tac x = "a" in allE)
-apply simp
-apply (erule_tac x = "aa" in allE)
-apply (erule_tac x = "b" in allE)
-apply (erule exE)
-apply (erule exE)
-apply (rule_tac x = " (s2,s2a) " in exI)
-apply (simp add: inp_is_act)
-(* a~: act B*)
-apply (simp add: inp_is_act)
-apply (erule_tac x = "a" in allE)
-apply simp
-apply (erule_tac x = "aa" in allE)
-apply (erule exE)
-apply (rule_tac x = " (s2,b) " in exI)
-apply simp
-
-(* a:inp B *)
-apply (case_tac "a:act A")
-(* a:act A *)
-apply (erule_tac x = "a" in allE)
-apply (erule_tac x = "a" in allE)
-apply (simp add: inp_is_act)
-apply (frule_tac A1 = "A" in compat_commute [THEN iffD1])
-apply (drule inpAAactB_is_inpBoroutB)
-back
-apply assumption
-apply assumption
-apply simp
-apply (erule_tac x = "aa" in allE)
-apply (erule_tac x = "b" in allE)
-apply (erule exE)
-apply (erule exE)
-apply (rule_tac x = " (s2,s2a) " in exI)
-apply (simp add: inp_is_act)
-(* a~: act B*)
-apply (simp add: inp_is_act)
-apply (erule_tac x = "a" in allE)
-apply (erule_tac x = "a" in allE)
-apply simp
-apply (erule_tac x = "b" in allE)
-apply (erule exE)
-apply (rule_tac x = " (aa,s2) " in exI)
-apply simp
-done
+lemma input_enabled_par:
+  "[| compatible A B; input_enabled A; input_enabled B|]
+        ==> input_enabled (A\<parallel>B)"
+  apply (unfold input_enabled_def)
+  apply (simp add: Let_def inputs_of_par trans_of_par)
+  apply (tactic "safe_tac (Context.raw_transfer @{theory} @{theory_context Fun})")
+  apply (simp add: inp_is_act)
+  prefer 2
+  apply (simp add: inp_is_act)
+  (* a: inp A *)
+  apply (case_tac "a:act B")
+  (* a:act B *)
+  apply (erule_tac x = "a" in allE)
+  apply simp
+  apply (drule inpAAactB_is_inpBoroutB)
+  apply assumption
+  apply assumption
+  apply (erule_tac x = "a" in allE)
+  apply simp
+  apply (erule_tac x = "aa" in allE)
+  apply (erule_tac x = "b" in allE)
+  apply (erule exE)
+  apply (erule exE)
+  apply (rule_tac x = " (s2,s2a) " in exI)
+  apply (simp add: inp_is_act)
+  (* a~: act B*)
+  apply (simp add: inp_is_act)
+  apply (erule_tac x = "a" in allE)
+  apply simp
+  apply (erule_tac x = "aa" in allE)
+  apply (erule exE)
+  apply (rule_tac x = " (s2,b) " in exI)
+  apply simp
+  
+  (* a:inp B *)
+  apply (case_tac "a:act A")
+  (* a:act A *)
+  apply (erule_tac x = "a" in allE)
+  apply (erule_tac x = "a" in allE)
+  apply (simp add: inp_is_act)
+  apply (frule_tac A1 = "A" in compat_commute [THEN iffD1])
+  apply (drule inpAAactB_is_inpBoroutB)
+  back
+  apply assumption
+  apply assumption
+  apply simp
+  apply (erule_tac x = "aa" in allE)
+  apply (erule_tac x = "b" in allE)
+  apply (erule exE)
+  apply (erule exE)
+  apply (rule_tac x = " (s2,s2a) " in exI)
+  apply (simp add: inp_is_act)
+  (* a~: act B*)
+  apply (simp add: inp_is_act)
+  apply (erule_tac x = "a" in allE)
+  apply (erule_tac x = "a" in allE)
+  apply simp
+  apply (erule_tac x = "b" in allE)
+  apply (erule exE)
+  apply (rule_tac x = " (aa,s2) " in exI)
+  apply simp
+  done
 
 
 subsection "invariants"
 
 lemma invariantI:
-  "[| !!s. s:starts_of(A) ==> P(s);      
-      !!s t a. [|reachable A s; P(s)|] ==> (s,a,t): trans_of(A) --> P(t) |]  
+  "[| !!s. s:starts_of(A) ==> P(s);
+      !!s t a. [|reachable A s; P(s)|] ==> (s,a,t): trans_of(A) --> P(t) |]
    ==> invariant A P"
-apply (unfold invariant_def)
-apply (rule allI)
-apply (rule impI)
-apply (rule_tac x = "s" in reachable.induct)
-apply assumption
-apply blast
-apply blast
-done
+  apply (unfold invariant_def)
+  apply (rule allI)
+  apply (rule impI)
+  apply (rule_tac x = "s" in reachable.induct)
+  apply assumption
+  apply blast
+  apply blast
+  done
 
 lemma invariantI1:
- "[| !!s. s : starts_of(A) ==> P(s);  
-     !!s t a. reachable A s ==> P(s) --> (s,a,t):trans_of(A) --> P(t)  
+ "[| !!s. s : starts_of(A) ==> P(s);
+     !!s t a. reachable A s ==> P(s) --> (s,a,t):trans_of(A) --> P(t)
   |] ==> invariant A P"
   apply (blast intro: invariantI)
   done
 
 lemma invariantE: "[| invariant A P; reachable A s |] ==> P(s)"

 
 
 lemmas reachable_0 = reachable.reachable_0
   and reachable_n = reachable.reachable_n
 
-lemma cancel_restrict_a: "starts_of(restrict ioa acts) = starts_of(ioa) &      
+lemma cancel_restrict_a: "starts_of(restrict ioa acts) = starts_of(ioa) &
           trans_of(restrict ioa acts) = trans_of(ioa)"
-apply (simp add: restrict_def ioa_projections)
-done
+  by (simp add: restrict_def ioa_projections)
 
 lemma cancel_restrict_b: "reachable (restrict ioa acts) s = reachable ioa s"
-apply (rule iffI)
-apply (erule reachable.induct)
-apply (simp add: cancel_restrict_a reachable_0)
-apply (erule reachable_n)
-apply (simp add: cancel_restrict_a)
-(* <--  *)
-apply (erule reachable.induct)
-apply (rule reachable_0)
-apply (simp add: cancel_restrict_a)
-apply (erule reachable_n)
-apply (simp add: cancel_restrict_a)
-done
+  apply (rule iffI)
+  apply (erule reachable.induct)
+  apply (simp add: cancel_restrict_a reachable_0)
+  apply (erule reachable_n)
+  apply (simp add: cancel_restrict_a)
+  (* <--  *)
+  apply (erule reachable.induct)
+  apply (rule reachable_0)
+  apply (simp add: cancel_restrict_a)
+  apply (erule reachable_n)
+  apply (simp add: cancel_restrict_a)
+  done
 
 lemma acts_restrict: "act (restrict A acts) = act A"
-apply (simp (no_asm) add: actions_def asig_internals_def
-  asig_outputs_def asig_inputs_def externals_def asig_of_def restrict_def restrict_asig_def)
-apply auto
-done
-
-lemma cancel_restrict: "starts_of(restrict ioa acts) = starts_of(ioa) &      
-          trans_of(restrict ioa acts) = trans_of(ioa) &  
-          reachable (restrict ioa acts) s = reachable ioa s &  
+  apply (simp (no_asm) add: actions_def asig_internals_def
+    asig_outputs_def asig_inputs_def externals_def asig_of_def restrict_def restrict_asig_def)
+  apply auto
+  done
+
+lemma cancel_restrict: "starts_of(restrict ioa acts) = starts_of(ioa) &
+          trans_of(restrict ioa acts) = trans_of(ioa) &
+          reachable (restrict ioa acts) s = reachable ioa s &
           act (restrict A acts) = act A"
-  apply (simp (no_asm) add: cancel_restrict_a cancel_restrict_b acts_restrict)
-  done
+  by (simp add: cancel_restrict_a cancel_restrict_b acts_restrict)
 
 
 subsection "rename"
 
 lemma trans_rename: "s \<midarrow>a\<midarrow>(rename C f)\<rightarrow> t ==> (? x. Some(x) = f(a) & s \<midarrow>x\<midarrow>C\<rightarrow> t)"
-apply (simp add: Let_def rename_def trans_of_def)
-done
+  by (simp add: Let_def rename_def trans_of_def)
 
 
 lemma reachable_rename: "[| reachable (rename C g) s |] ==> reachable C s"
-apply (erule reachable.induct)
-apply (rule reachable_0)
-apply (simp add: rename_def ioa_projections)
-apply (drule trans_rename)
-apply (erule exE)
-apply (erule conjE)
-apply (erule reachable_n)
-apply assumption
-done
+  apply (erule reachable.induct)
+  apply (rule reachable_0)
+  apply (simp add: rename_def ioa_projections)
+  apply (drule trans_rename)
+  apply (erule exE)
+  apply (erule conjE)
+  apply (erule reachable_n)
+  apply assumption
+  done
 
 
 subsection "trans_of(A\<parallel>B)"
 
-
-lemma trans_A_proj: "[|(s,a,t):trans_of (A\<parallel>B); a:act A|]  
+lemma trans_A_proj: "[|(s,a,t):trans_of (A\<parallel>B); a:act A|]
               ==> (fst s,a,fst t):trans_of A"
-apply (simp add: Let_def par_def trans_of_def)
-done
-
-lemma trans_B_proj: "[|(s,a,t):trans_of (A\<parallel>B); a:act B|]  
+  by (simp add: Let_def par_def trans_of_def)
+
+lemma trans_B_proj: "[|(s,a,t):trans_of (A\<parallel>B); a:act B|]
               ==> (snd s,a,snd t):trans_of B"
-apply (simp add: Let_def par_def trans_of_def)
-done
-
-lemma trans_A_proj2: "[|(s,a,t):trans_of (A\<parallel>B); a~:act A|] 
+  by (simp add: Let_def par_def trans_of_def)
+
+lemma trans_A_proj2: "[|(s,a,t):trans_of (A\<parallel>B); a~:act A|]
               ==> fst s = fst t"
-apply (simp add: Let_def par_def trans_of_def)
-done
-
-lemma trans_B_proj2: "[|(s,a,t):trans_of (A\<parallel>B); a~:act B|] 
+  by (simp add: Let_def par_def trans_of_def)
+
+lemma trans_B_proj2: "[|(s,a,t):trans_of (A\<parallel>B); a~:act B|]
               ==> snd s = snd t"
-apply (simp add: Let_def par_def trans_of_def)
-done
-
-lemma trans_AB_proj: "(s,a,t):trans_of (A\<parallel>B)  
+  by (simp add: Let_def par_def trans_of_def)
+
+lemma trans_AB_proj: "(s,a,t):trans_of (A\<parallel>B)
                ==> a :act A | a :act B"
-apply (simp add: Let_def par_def trans_of_def)
-done
-
-lemma trans_AB: "[|a:act A;a:act B; 
-       (fst s,a,fst t):trans_of A;(snd s,a,snd t):trans_of B|] 
+  by (simp add: Let_def par_def trans_of_def)
+
+lemma trans_AB: "[|a:act A;a:act B;
+       (fst s,a,fst t):trans_of A;(snd s,a,snd t):trans_of B|]
    ==> (s,a,t):trans_of (A\<parallel>B)"
-apply (simp add: Let_def par_def trans_of_def)
-done
-
-lemma trans_A_notB: "[|a:act A;a~:act B; 
-       (fst s,a,fst t):trans_of A;snd s=snd t|] 
+  by (simp add: Let_def par_def trans_of_def)
+
+lemma trans_A_notB: "[|a:act A;a~:act B;
+       (fst s,a,fst t):trans_of A;snd s=snd t|]
    ==> (s,a,t):trans_of (A\<parallel>B)"
-apply (simp add: Let_def par_def trans_of_def)
-done
-
-lemma trans_notA_B: "[|a~:act A;a:act B; 
-       (snd s,a,snd t):trans_of B;fst s=fst t|] 
+  by (simp add: Let_def par_def trans_of_def)
+
+lemma trans_notA_B: "[|a~:act A;a:act B;
+       (snd s,a,snd t):trans_of B;fst s=fst t|]
    ==> (s,a,t):trans_of (A\<parallel>B)"
-apply (simp add: Let_def par_def trans_of_def)
-done
+  by (simp add: Let_def par_def trans_of_def)
 
 lemmas trans_of_defs1 = trans_AB trans_A_notB trans_notA_B
   and trans_of_defs2 = trans_A_proj trans_B_proj trans_A_proj2 trans_B_proj2 trans_AB_proj
 
 
-lemma trans_of_par4: 
-"((s,a,t) : trans_of(A \<parallel> B \<parallel> C \<parallel> D)) =                                     
-  ((a:actions(asig_of(A)) | a:actions(asig_of(B)) | a:actions(asig_of(C)) |   
-    a:actions(asig_of(D))) &                                                  
-   (if a:actions(asig_of(A)) then (fst(s),a,fst(t)):trans_of(A)               
-    else fst t=fst s) &                                                       
-   (if a:actions(asig_of(B)) then (fst(snd(s)),a,fst(snd(t))):trans_of(B)     
-    else fst(snd(t))=fst(snd(s))) &                                           
-   (if a:actions(asig_of(C)) then                                             
-      (fst(snd(snd(s))),a,fst(snd(snd(t)))):trans_of(C)                       
-    else fst(snd(snd(t)))=fst(snd(snd(s)))) &                                 
-   (if a:actions(asig_of(D)) then                                             
-      (snd(snd(snd(s))),a,snd(snd(snd(t)))):trans_of(D)                       
+lemma trans_of_par4:
+"((s,a,t) : trans_of(A \<parallel> B \<parallel> C \<parallel> D)) =
+  ((a:actions(asig_of(A)) | a:actions(asig_of(B)) | a:actions(asig_of(C)) |
+    a:actions(asig_of(D))) &
+   (if a:actions(asig_of(A)) then (fst(s),a,fst(t)):trans_of(A)
+    else fst t=fst s) &
+   (if a:actions(asig_of(B)) then (fst(snd(s)),a,fst(snd(t))):trans_of(B)
+    else fst(snd(t))=fst(snd(s))) &
+   (if a:actions(asig_of(C)) then
+      (fst(snd(snd(s))),a,fst(snd(snd(t)))):trans_of(C)
+    else fst(snd(snd(t)))=fst(snd(snd(s)))) &
+   (if a:actions(asig_of(D)) then
+      (snd(snd(snd(s))),a,snd(snd(snd(t)))):trans_of(D)
     else snd(snd(snd(t)))=snd(snd(snd(s)))))"
-  apply (simp (no_asm) add: par_def actions_asig_comp prod_eq_iff Let_def ioa_projections)
-  done
+  by (simp add: par_def actions_asig_comp prod_eq_iff Let_def ioa_projections)
 
 
 subsection "proof obligation generator for IOA requirements"
 
 (* without assumptions on A and B because is_trans_of is also incorporated in \<parallel>def *)
 lemma is_trans_of_par: "is_trans_of (A\<parallel>B)"
-apply (unfold is_trans_of_def)
-apply (simp add: Let_def actions_of_par trans_of_par)
-done
-
-lemma is_trans_of_restrict: 
-"is_trans_of A ==> is_trans_of (restrict A acts)"
-apply (unfold is_trans_of_def)
-apply (simp add: cancel_restrict acts_restrict)
-done
-
-lemma is_trans_of_rename: 
-"is_trans_of A ==> is_trans_of (rename A f)"
-apply (unfold is_trans_of_def restrict_def restrict_asig_def)
-apply (simp add: Let_def actions_def trans_of_def asig_internals_def
-  asig_outputs_def asig_inputs_def externals_def asig_of_def rename_def rename_set_def)
-apply blast
-done
-
-lemma is_asig_of_par: "[| is_asig_of A; is_asig_of B; compatible A B|]   
+  by (simp add: is_trans_of_def Let_def actions_of_par trans_of_par)
+
+lemma is_trans_of_restrict: "is_trans_of A ==> is_trans_of (restrict A acts)"
+  by (simp add: is_trans_of_def cancel_restrict acts_restrict)
+
+lemma is_trans_of_rename: "is_trans_of A ==> is_trans_of (rename A f)"
+  apply (unfold is_trans_of_def restrict_def restrict_asig_def)
+  apply (simp add: Let_def actions_def trans_of_def asig_internals_def
+    asig_outputs_def asig_inputs_def externals_def asig_of_def rename_def rename_set_def)
+  apply blast
+  done
+
+lemma is_asig_of_par: "[| is_asig_of A; is_asig_of B; compatible A B|]
           ==> is_asig_of (A\<parallel>B)"
-apply (simp add: is_asig_of_def asig_of_par asig_comp_def compatible_def
-  asig_internals_def asig_outputs_def asig_inputs_def actions_def is_asig_def)
-apply (simp add: asig_of_def)
-apply auto
-done
-
-lemma is_asig_of_restrict: 
-"is_asig_of A ==> is_asig_of (restrict A f)"
-apply (unfold is_asig_of_def is_asig_def asig_of_def restrict_def restrict_asig_def 
-           asig_internals_def asig_outputs_def asig_inputs_def externals_def o_def)
-apply simp
-apply auto
-done
+  apply (simp add: is_asig_of_def asig_of_par asig_comp_def compatible_def
+    asig_internals_def asig_outputs_def asig_inputs_def actions_def is_asig_def)
+  apply (simp add: asig_of_def)
+  apply auto
+  done
+
+lemma is_asig_of_restrict: "is_asig_of A ==> is_asig_of (restrict A f)"
+  apply (unfold is_asig_of_def is_asig_def asig_of_def restrict_def restrict_asig_def
+             asig_internals_def asig_outputs_def asig_inputs_def externals_def o_def)
+  apply simp
+  apply auto
+  done
 
 lemma is_asig_of_rename: "is_asig_of A ==> is_asig_of (rename A f)"
-apply (simp add: is_asig_of_def rename_def rename_set_def asig_internals_def
-  asig_outputs_def asig_inputs_def actions_def is_asig_def asig_of_def)
-apply auto
-apply (drule_tac [!] s = "Some _" in sym)
-apply auto
-done
+  apply (simp add: is_asig_of_def rename_def rename_set_def asig_internals_def
+    asig_outputs_def asig_inputs_def actions_def is_asig_def asig_of_def)
+  apply auto
+  apply (drule_tac [!] s = "Some _" in sym)
+  apply auto
+  done
 
 lemmas [simp] = is_asig_of_par is_asig_of_restrict
   is_asig_of_rename is_trans_of_par is_trans_of_restrict is_trans_of_rename
 
 
-lemma compatible_par: 
-"[|compatible A B; compatible A C |]==> compatible A (B\<parallel>C)"
-apply (unfold compatible_def)
-apply (simp add: internals_of_par outputs_of_par actions_of_par)
-apply auto
-done
+lemma compatible_par: "[|compatible A B; compatible A C |]==> compatible A (B\<parallel>C)"
+  apply (unfold compatible_def)
+  apply (simp add: internals_of_par outputs_of_par actions_of_par)
+  apply auto
+  done
 
 (*  better derive by previous one and compat_commute *)
-lemma compatible_par2: 
-"[|compatible A C; compatible B C |]==> compatible (A\<parallel>B) C"
-apply (unfold compatible_def)
-apply (simp add: internals_of_par outputs_of_par actions_of_par)
-apply auto
-done
-
-lemma compatible_restrict: 
-"[| compatible A B; (ext B - S) Int ext A = {}|]  
-      ==> compatible A (restrict B S)"
-apply (unfold compatible_def)
-apply (simp add: ioa_triple_proj asig_triple_proj externals_def
-  restrict_def restrict_asig_def actions_def)
-apply auto
-done
-
+lemma compatible_par2: "[|compatible A C; compatible B C |]==> compatible (A\<parallel>B) C"
+  apply (unfold compatible_def)
+  apply (simp add: internals_of_par outputs_of_par actions_of_par)
+  apply auto
+  done
+
+lemma compatible_restrict:
+  "[| compatible A B; (ext B - S) Int ext A = {}|]
+        ==> compatible A (restrict B S)"
+  apply (unfold compatible_def)
+  apply (simp add: ioa_triple_proj asig_triple_proj externals_def
+    restrict_def restrict_asig_def actions_def)
+  apply auto
+  done
 
 declare split_paired_Ex [simp]
 
 end