src/HOL/Nominal/Examples/SOS.thy

 (* "$Id$" *)
-(*                                                   *)
-(* Formalisation of some typical SOS-proofs          *)
-(*                                                   *) 
-(* This work arose from challenge suggested by Adam  *)
-(* Chlipala suggested on the POPLmark mailing list.  *)
-(*                                                   *) 
-(* We thank Nick Benton for helping us with the      *) 
-(* termination-proof for evaluation.                 *)
-(*                                                   *)
-(* The formalisation was done by Julien Narboux and  *)
-(* Christian Urban.                                  *)
+(*                                                     *)
+(* Formalisation of some typical SOS-proofs.           *)
+(*                                                     *) 
+(* This work arose from a challenge suggested by Adam  *)
+(* Chlipala on the POPLmark mailing list.              *)
+(*                                                     *) 
+(* We thank Nick Benton for helping us with the        *) 
+(* termination-proof for evaluation.                   *)
+(*                                                     *)
+(* The formalisation was done by Julien Narboux and    *)
+(* Christian Urban.                                    *)
 
 theory SOS
   imports "../Nominal"
 begin
 
 atom_decl name
 
-nominal_datatype data = 
-    DNat
-  | DProd "data" "data"
-  | DSum "data" "data"
-
+text {* types and terms *}
 nominal_datatype ty = 
-    Data "data"
+    TVar "nat"
   | Arrow "ty" "ty" ("_\<rightarrow>_" [100,100] 100)
 
 nominal_datatype trm = 
     Var "name"
   | Lam "\<guillemotleft>name\<guillemotright>trm" ("Lam [_]._" [100,100] 100)
   | App "trm" "trm"
-  | Const "nat"
-  | Pr "trm" "trm"
-  | Fst "trm"
-  | Snd "trm"
-  | InL "trm"
-  | InR "trm"
-  | Case "trm" "\<guillemotleft>name\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" ("Case _ of inl _ \<rightarrow> _ | inr _ \<rightarrow> _" [100,100,100,100,100] 100)
-
-lemma in_eqvt[eqvt]:
-  fixes pi::"name prm"
-  and   x::"'a::pt_name"
-  assumes "x\<in>X"
-  shows "pi\<bullet>x \<in> pi\<bullet>X"
-  using assms by (perm_simp add: pt_set_bij1a[OF pt_name_inst, OF at_name_inst])
-
-lemma perm_data[simp]: 
-  fixes D::"data"
-  and   pi::"name prm"
-  shows "pi\<bullet>D = D"
-  by (induct D rule: data.weak_induct) (simp_all)
-
-lemma perm_ty[simp]: 
-  fixes T::"ty"
-  and   pi::"name prm"
-  shows "pi\<bullet>T = T"
-  by (induct T rule: ty.weak_induct) (simp_all)
-
-lemma fresh_ty[simp]:
+
+lemma fresh_ty:
   fixes x::"name" 
   and   T::"ty"
   shows "x\<sharp>T"
-  by (simp add: fresh_def supp_def)
-
-text {* substitution *}
-
+by (induct T rule: ty.weak_induct)
+   (auto simp add: fresh_nat)
+
+text {* Parallel and single substitution. *}
 fun
   lookup :: "(name\<times>trm) list \<Rightarrow> name \<Rightarrow> trm"   
 where
   "lookup [] x        = Var x"
 | "lookup ((y,e)#\<theta>) x = (if x=y then e else lookup \<theta> x)"
 
-lemma lookup_eqvt:
+lemma lookup_eqvt[eqvt]:
   fixes pi::"name prm"
   and   \<theta>::"(name\<times>trm) list"
   and   X::"name"
   shows "pi\<bullet>(lookup \<theta> X) = lookup (pi\<bullet>\<theta>) (pi\<bullet>X)"
-by (induct \<theta>, auto simp add: perm_bij)
+by (induct \<theta>) (auto simp add: eqvts)
 
 lemma lookup_fresh:
   fixes z::"name"
-  assumes "z\<sharp>\<theta>" and "z\<sharp>x"
+  assumes a: "z\<sharp>\<theta>" and b: "z\<sharp>x"
   shows "z \<sharp>lookup \<theta> x"
-using assms 
+using a b
 by (induct rule: lookup.induct) (auto simp add: fresh_list_cons)
 
 lemma lookup_fresh':
   assumes "z\<sharp>\<theta>"
   shows "lookup \<theta> z = Var z"
 using assms 
 by (induct rule: lookup.induct)
    (auto simp add: fresh_list_cons fresh_prod fresh_atm)
 
-text {* Parallel Substitution *}
-
+(* parallel substitution *)
 consts
   psubst :: "(name\<times>trm) list \<Rightarrow> trm \<Rightarrow> trm"  ("_<_>" [95,95] 105)
  
 nominal_primrec
   "\<theta><(Var x)> = (lookup \<theta> x)"
   "\<theta><(App e\<^isub>1 e\<^isub>2)> = App (\<theta><e\<^isub>1>) (\<theta><e\<^isub>2>)"
   "x\<sharp>\<theta> \<Longrightarrow> \<theta><(Lam [x].e)> = Lam [x].(\<theta><e>)"
-  "\<theta><(Const n)> = Const n"
-  "\<theta><(Pr e\<^isub>1 e\<^isub>2)> = Pr (\<theta><e\<^isub>1>) (\<theta><e\<^isub>2>)"
-  "\<theta><(Fst e)> = Fst (\<theta><e>)"
-  "\<theta><(Snd e)> = Snd (\<theta><e>)"
-  "\<theta><(InL e)> = InL (\<theta><e>)"
-  "\<theta><(InR e)> = InR (\<theta><e>)"
-  "\<lbrakk>y\<noteq>x; x\<sharp>(e,e\<^isub>2,\<theta>); y\<sharp>(e,e\<^isub>1,\<theta>)\<rbrakk> 
-   \<Longrightarrow> \<theta><Case e of inl x \<rightarrow> e\<^isub>1 | inr y \<rightarrow> e\<^isub>2> = (Case (\<theta><e>) of inl x \<rightarrow> (\<theta><e\<^isub>1>) | inr y \<rightarrow> (\<theta><e\<^isub>2>))"
-apply(finite_guess add: lookup_eqvt)+
+apply(finite_guess)+
 apply(rule TrueI)+
 apply(simp add: abs_fresh)+
-apply(fresh_guess add: fs_name1 lookup_eqvt)+
+apply(fresh_guess)+
 done
 
 lemma psubst_eqvt[eqvt]:
   fixes pi::"name prm" 
   and   t::"trm"

   shows "z\<sharp>(\<theta><t>)"
 using assms
 by (nominal_induct t avoiding: z \<theta> t rule: trm.induct)
    (auto simp add: abs_fresh lookup_fresh)
 
+lemma psubst_empty[simp]:
+  shows "[]<t> = t"
+  by (nominal_induct t rule: trm.induct) 
+     (auto simp add: fresh_list_nil)
+
+(* Single substitution *)
 abbreviation 
- subst :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm" ("_[_::=_]" [100,100,100] 100)
-  where "t[x::=t']  \<equiv> ([(x,t')])<t>" 
+  subst :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm" ("_[_::=_]" [100,100,100] 100)
+where 
+  "t[x::=t']  \<equiv> ([(x,t')])<t>" 
 
 lemma subst[simp]:
   shows "(Var x)[y::=t'] = (if x=y then t' else (Var x))"
   and   "(App t\<^isub>1 t\<^isub>2)[y::=t'] = App (t\<^isub>1[y::=t']) (t\<^isub>2[y::=t'])"
   and   "x\<sharp>(y,t') \<Longrightarrow> (Lam [x].t)[y::=t'] = Lam [x].(t[y::=t'])"
-  and   "(Const n)[y::=t'] = Const n"
-  and   "(Pr e\<^isub>1 e\<^isub>2)[y::=t'] = Pr (e\<^isub>1[y::=t']) (e\<^isub>2[y::=t'])"
-  and   "(Fst e)[y::=t'] = Fst (e[y::=t'])"
-  and   "(Snd e)[y::=t'] = Snd (e[y::=t'])"
-  and   "(InL e)[y::=t'] = InL (e[y::=t'])"
-  and   "(InR e)[y::=t'] = InR (e[y::=t'])"
-  and   "\<lbrakk>z\<noteq>x; x\<sharp>(y,e,e\<^isub>2,t'); z\<sharp>(y,e,e\<^isub>1,t')\<rbrakk> 
-         \<Longrightarrow> (Case e of inl x \<rightarrow> e\<^isub>1 | inr z \<rightarrow> e\<^isub>2)[y::=t'] =
-                   (Case (e[y::=t']) of inl x \<rightarrow> (e\<^isub>1[y::=t']) | inr z \<rightarrow> (e\<^isub>2[y::=t']))"
 by (simp_all add: fresh_list_cons fresh_list_nil)
-
-lemma subst_eqvt[eqvt]:
-  fixes pi::"name prm" 
-  and   t::"trm"
-  shows "pi\<bullet>(t[x::=t']) = (pi\<bullet>t)[(pi\<bullet>x)::=(pi\<bullet>t')]"
-  by (nominal_induct t avoiding: x t' rule: trm.induct)
-     (perm_simp add: fresh_bij)+
 
 lemma fresh_subst:
   fixes z::"name"
   and   t\<^isub>1::"trm"
   and   t2::"trm"

 using assms 
 by (nominal_induct t\<^isub>1 avoiding: z y t\<^isub>2 rule: trm.induct)
    (auto simp add: abs_fresh fresh_atm)
 
 lemma fresh_subst':
-  fixes z::"name"
-  and   t\<^isub>1::"trm"
-  and   t2::"trm"
-  assumes "z\<sharp>[y].t\<^isub>1" and "z\<sharp>t\<^isub>2"
-  shows "z\<sharp>t\<^isub>1[y::=t\<^isub>2]"
-using assms 
-by (nominal_induct t\<^isub>1 avoiding: y t\<^isub>2 z  rule: trm.induct)
-   (auto simp add: abs_fresh fresh_nat fresh_atm)
+  assumes "x\<sharp>e'"
+  shows "x\<sharp>e[x::=e']"
+  using assms 
+by (nominal_induct e avoiding: x e' rule: trm.induct)
+   (auto simp add: fresh_atm abs_fresh fresh_nat) 
 
 lemma forget: 
   fixes x::"name"
   and   L::"trm"
-  assumes "x\<sharp>L" 
-  shows "L[x::=P] = L"
+  assumes "x\<sharp>e" 
+  shows "e[x::=e'] = e"
   using assms
-  by (nominal_induct L avoiding: x P rule: trm.induct)
+  by (nominal_induct e avoiding: x e' rule: trm.induct)
      (auto simp add: fresh_atm abs_fresh)
 
-lemma psubst_empty[simp]:
-  shows "[]<t> = t"
-  by (nominal_induct t rule: trm.induct, auto simp add:fresh_list_nil)
-
 lemma psubst_subst_psubst:
-assumes h:"x\<sharp>\<theta>"
-shows "\<theta><e>[x::=e'] = ((x,e')#\<theta>)<e>"
-using h
+  assumes h:"x\<sharp>\<theta>"
+  shows "\<theta><e>[x::=e'] = ((x,e')#\<theta>)<e>"
+  using h
 apply(nominal_induct e avoiding: \<theta> x e' rule: trm.induct)
-apply(auto simp add: fresh_list_cons fresh_atm forget lookup_fresh lookup_fresh' fresh_psubst)
+apply(auto simp add: fresh_list_cons fresh_atm forget lookup_fresh lookup_fresh')
 done
 
-lemma fresh_subst_fresh:
-    assumes "a\<sharp>e"
-    shows "a\<sharp>t[a::=e]"
-using assms 
-by (nominal_induct t avoiding: a e rule: trm.induct)
-   (auto simp add: fresh_atm abs_fresh fresh_nat) 
-
-text {* Typing-Judgements *}
+text {* Typing Judgements *}
 
 inductive
-  valid :: "(name \<times> 'a::pt_name) list \<Rightarrow> bool"
-where
-    v_nil[intro]:  "valid []"
-  | v_cons[intro]: "\<lbrakk>valid \<Gamma>;x\<sharp>\<Gamma>\<rbrakk> \<Longrightarrow> valid ((x,T)#\<Gamma>)"
+  valid :: "(name\<times>ty) list \<Rightarrow> bool"
+where
+  v_nil[intro]:  "valid []"
+| v_cons[intro]: "\<lbrakk>valid \<Gamma>;x\<sharp>\<Gamma>\<rbrakk> \<Longrightarrow> valid ((x,T)#\<Gamma>)"
 
 equivariance valid 
 
-inductive_cases  
-  valid_cons_inv_auto[elim]:"valid ((x,T)#\<Gamma>)"
-
-abbreviation
-  "sub" :: "(name\<times>ty) list \<Rightarrow> (name\<times>ty) list \<Rightarrow> bool" ("_ \<subseteq> _" [55,55] 55)
-where
-  "\<Gamma>\<^isub>1 \<subseteq> \<Gamma>\<^isub>2 \<equiv> \<forall>x T. (x,T)\<in>set \<Gamma>\<^isub>1 \<longrightarrow> (x,T)\<in>set \<Gamma>\<^isub>2"
+inductive_cases
+  valid_cons_inv_auto[elim]: "valid ((x,T)#\<Gamma>)"
 
 lemma type_unicity_in_context:
   assumes asm1: "(x,t\<^isub>2) \<in> set ((x,t\<^isub>1)#\<Gamma>)" 
   and     asm2: "valid ((x,t\<^isub>1)#\<Gamma>)"
   shows "t\<^isub>1=t\<^isub>2"

   then have "(x,t\<^isub>2) = (x,t\<^isub>1)" using asm1 by auto
   then show "t\<^isub>1 = t\<^isub>2" by auto
 qed
 
 lemma case_distinction_on_context:
-  fixes \<Gamma>::"(name \<times> ty) list"
+  fixes \<Gamma>::"(name\<times>ty) list"
   assumes asm1: "valid ((m,t)#\<Gamma>)" 
   and     asm2: "(n,U) \<in> set ((m,T)#\<Gamma>)"
   shows "(n,U) = (m,T) \<or> ((n,U) \<in> set \<Gamma> \<and> n \<noteq> m)"
 proof -
-from asm2 have "(n,U) \<in> set [(m,T)] \<or> (n,U) \<in> set \<Gamma>" by auto
-moreover
-{ assume eq: "m=n"
-  assume "(n,U) \<in> set \<Gamma>" 
-  then have "\<not> n\<sharp>\<Gamma>" 
-    by (induct \<Gamma>) (auto simp add: fresh_list_cons fresh_prod fresh_atm)
-  moreover have "m\<sharp>\<Gamma>" using asm1 by auto
-  ultimately have False using eq by auto
-}
-ultimately show ?thesis by auto
+  from asm2 have "(n,U) \<in> set [(m,T)] \<or> (n,U) \<in> set \<Gamma>" by auto
+  moreover
+  { assume eq: "m=n"
+    assume "(n,U) \<in> set \<Gamma>" 
+    then have "\<not> n\<sharp>\<Gamma>" 
+      by (induct \<Gamma>) (auto simp add: fresh_list_cons fresh_prod fresh_atm)
+    moreover have "m\<sharp>\<Gamma>" using asm1 by auto
+    ultimately have False using eq by auto
+  }
+  ultimately show ?thesis by auto
 qed
 
 inductive
   typing :: "(name\<times>ty) list\<Rightarrow>trm\<Rightarrow>ty\<Rightarrow>bool" ("_ \<turnstile> _ : _" [60,60,60] 60) 
 where
-  t_Var[intro]:   "\<lbrakk>valid \<Gamma>; (x,T)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
-| t_App[intro]:   "\<lbrakk>\<Gamma> \<turnstile> e\<^isub>1 : T\<^isub>1\<rightarrow>T\<^isub>2; \<Gamma> \<turnstile> e\<^isub>2 : T\<^isub>1\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> App e\<^isub>1 e\<^isub>2 : T\<^isub>2"
+  t_Var[intro]:   "\<lbrakk>valid \<Gamma>; (x,T)\<in>set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
+| t_App[intro]:   "\<lbrakk>\<Gamma> \<turnstile> e\<^isub>1 : T\<^isub>1\<rightarrow>T\<^isub>2; \<Gamma> \<turnstile> e\<^isub>2 : T\<^isub>1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App e\<^isub>1 e\<^isub>2 : T\<^isub>2"
 | t_Lam[intro]:   "\<lbrakk>x\<sharp>\<Gamma>; (x,T\<^isub>1)#\<Gamma> \<turnstile> e : T\<^isub>2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].e : T\<^isub>1\<rightarrow>T\<^isub>2"
-| t_Const[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile> Const n : Data(DNat)"
-| t_Pr[intro]:    "\<lbrakk>\<Gamma> \<turnstile> e\<^isub>1 : Data(S\<^isub>1); \<Gamma> \<turnstile> e\<^isub>2 : Data(S\<^isub>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Pr e\<^isub>1 e\<^isub>2 : Data (DProd S\<^isub>1 S\<^isub>2)"
-| t_Fst[intro]:   "\<lbrakk>\<Gamma> \<turnstile> e : Data(DProd S\<^isub>1 S\<^isub>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Fst e : Data(S\<^isub>1)"
-| t_Snd[intro]:   "\<lbrakk>\<Gamma> \<turnstile> e : Data(DProd S\<^isub>1 S\<^isub>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Snd e : Data(S\<^isub>2)"
-| t_InL[intro]:   "\<lbrakk>\<Gamma> \<turnstile> e : Data(S\<^isub>1)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> InL e : Data(DSum S\<^isub>1 S\<^isub>2)"
-| t_InR[intro]:   "\<lbrakk>\<Gamma> \<turnstile> e : Data(S\<^isub>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> InR e : Data(DSum S\<^isub>1 S\<^isub>2)"
-| t_Case[intro]:  "\<lbrakk>x\<^isub>1\<sharp>(\<Gamma>,e,e\<^isub>2,x\<^isub>2); x\<^isub>2\<sharp>(\<Gamma>,e,e\<^isub>1,x\<^isub>1); \<Gamma> \<turnstile> e: Data(DSum S\<^isub>1 S\<^isub>2); 
-                   (x\<^isub>1,Data(S\<^isub>1))#\<Gamma> \<turnstile> e\<^isub>1 : T; (x\<^isub>2,Data(S\<^isub>2))#\<Gamma> \<turnstile> e\<^isub>2 : T\<rbrakk> 
-                   \<Longrightarrow> \<Gamma> \<turnstile> (Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2) : T"
 
 equivariance typing
 
 nominal_inductive typing
-  by (simp_all add: abs_fresh fresh_prod fresh_atm)
-
-lemmas typing_eqvt' = typing.eqvt[simplified]
+  by (simp_all add: abs_fresh fresh_ty)
 
 lemma typing_implies_valid:
-  assumes "\<Gamma> \<turnstile> t : T"
+  assumes a: "\<Gamma> \<turnstile> t : T"
   shows "valid \<Gamma>"
-  using assms
+  using a
   by (induct) (auto)
 
-declare trm.inject [simp add]
-declare ty.inject  [simp add]
-declare data.inject [simp add]
-
-
-inductive_cases typing_inv_auto[elim]: 
-  "\<Gamma> \<turnstile> Lam [x].t : T"
-  "\<Gamma> \<turnstile> Var x : T"
-  "\<Gamma> \<turnstile> App x y : T"
-  "\<Gamma> \<turnstile> Const n : T"
-  "\<Gamma> \<turnstile> Fst x : T"
-  "\<Gamma> \<turnstile> Snd x : T"
-  "\<Gamma> \<turnstile> InL x : T"
-  "\<Gamma> \<turnstile> InL x : Data (DSum T\<^isub>1 T2)"
-  "\<Gamma> \<turnstile> InR x : T"
-  "\<Gamma> \<turnstile> InR x : Data (DSum T\<^isub>1 T2)"
-  "\<Gamma> \<turnstile> Pr x y : T"
-  "\<Gamma> \<turnstile> Pr e\<^isub>1 e\<^isub>2 : Data (DProd \<sigma>1 \<sigma>\<^isub>2)"
-  "\<Gamma> \<turnstile> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 : T"
-
-declare trm.inject [simp del]
-declare ty.inject [simp del]
-declare data.inject [simp del]
-
-lemma t_Lam_elim[elim]: 
+lemma t_Lam_elim: 
   assumes a1:"\<Gamma> \<turnstile> Lam [x].t : T" 
   and     a2: "x\<sharp>\<Gamma>"
   obtains T\<^isub>1 and T\<^isub>2 where "(x,T\<^isub>1)#\<Gamma> \<turnstile> t : T\<^isub>2" and "T=T\<^isub>1\<rightarrow>T\<^isub>2"
-proof -
-  from a1 obtain x' t' T\<^isub>1 T\<^isub>2 
-    where b1: "x'\<sharp>\<Gamma>" and b2: "(x',T\<^isub>1)#\<Gamma> \<turnstile> t' : T\<^isub>2" and b3: "[x'].t' = [x].t" and b4: "T=T\<^isub>1\<rightarrow>T\<^isub>2"
-    by auto
-  obtain c::"name" where "c\<sharp>(\<Gamma>,x,x',t,t')" by (erule exists_fresh[OF fs_name1])
-  then have fs: "c\<sharp>\<Gamma>" "c\<noteq>x" "c\<noteq>x'" "c\<sharp>t" "c\<sharp>t'" by (simp_all add: fresh_atm[symmetric]) 
-  then have b5: "[(x',c)]\<bullet>t'=[(x,c)]\<bullet>t" using b3 fs by (simp add: alpha_fresh)
-  have "([(x,c)]\<bullet>[(x',c)]\<bullet>((x',T\<^isub>1)#\<Gamma>)) \<turnstile> ([(x,c)]\<bullet>[(x',c)]\<bullet>t') : T\<^isub>2" using b2
-    by (simp only: typing_eqvt')
-  then have "(x,T\<^isub>1)#\<Gamma> \<turnstile> t : T\<^isub>2" using fs b1 a2 b5 by (perm_simp add: calc_atm)
-  then show ?thesis using prems b4 by simp
-qed
-
-lemma t_Case_elim[elim]: 
-  assumes "\<Gamma> \<turnstile> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 : T" and "x\<^isub>1\<sharp>\<Gamma>" and "x\<^isub>2\<sharp>\<Gamma>" 
-  obtains \<sigma>\<^isub>1 \<sigma>\<^isub>2 where "\<Gamma> \<turnstile> e : Data (DSum \<sigma>\<^isub>1 \<sigma>\<^isub>2)" 
-                  and "(x\<^isub>1, Data \<sigma>\<^isub>1)#\<Gamma> \<turnstile> e\<^isub>1 : T" 
-                  and "(x\<^isub>2, Data \<sigma>\<^isub>2)#\<Gamma> \<turnstile> e\<^isub>2 : T"
-proof -
-  have f:"x\<^isub>1\<sharp>\<Gamma>" "x\<^isub>2\<sharp>\<Gamma>" by fact+
-  have "\<Gamma> \<turnstile> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 : T" by fact
-  then obtain \<sigma>\<^isub>1 \<sigma>\<^isub>2 x\<^isub>1' x\<^isub>2' e\<^isub>1' e\<^isub>2' where 
-    h:"\<Gamma> \<turnstile> e : Data (DSum \<sigma>\<^isub>1 \<sigma>\<^isub>2)" and 
-    h1:"(x\<^isub>1',Data \<sigma>\<^isub>1)#\<Gamma> \<turnstile> e\<^isub>1' : T" and 
-    h2:"(x\<^isub>2',Data \<sigma>\<^isub>2)#\<Gamma> \<turnstile> e\<^isub>2' : T" and
-    e1:"[x\<^isub>1].e\<^isub>1=[x\<^isub>1'].e\<^isub>1'" and e2:"[x\<^isub>2].e\<^isub>2=[x\<^isub>2'].e\<^isub>2'"
-    by auto
-  obtain c::name where f':"c \<sharp> (x\<^isub>1,x\<^isub>1',e\<^isub>1,e\<^isub>1',\<Gamma>)" by (erule exists_fresh[OF fs_name1])
-  have e1':"[(x\<^isub>1,c)]\<bullet>e\<^isub>1 = [(x\<^isub>1',c)]\<bullet>e\<^isub>1'" using e1 f' by (auto simp add: alpha_fresh fresh_prod fresh_atm)
-  have "[(x\<^isub>1',c)]\<bullet>((x\<^isub>1',Data \<sigma>\<^isub>1)# \<Gamma>) \<turnstile> [(x\<^isub>1',c)]\<bullet>e\<^isub>1' : T" using h1 typing_eqvt' by blast
-  then have x:"(c,Data \<sigma>\<^isub>1)#( [(x\<^isub>1',c)]\<bullet>\<Gamma>) \<turnstile> [(x\<^isub>1',c)]\<bullet>e\<^isub>1': T" using f' 
-    by (auto simp add: fresh_atm calc_atm)
-  have "x\<^isub>1' \<sharp> \<Gamma>" using h1 typing_implies_valid by auto
-  then have "(c,Data \<sigma>\<^isub>1)#\<Gamma> \<turnstile> [(x\<^isub>1 ,c)]\<bullet>e\<^isub>1 : T" using f' x e1' by (auto simp add: perm_fresh_fresh)
-  then have "[(x\<^isub>1,c)]\<bullet>((c,Data \<sigma>\<^isub>1)#\<Gamma>) \<turnstile> [(x\<^isub>1,c)]\<bullet>[(x\<^isub>1 ,c)]\<bullet>e\<^isub>1 : T" using typing_eqvt' by blast 
-  then have "([(x\<^isub>1,c)]\<bullet>(c,Data \<sigma>\<^isub>1)) #\<Gamma> \<turnstile> [(x\<^isub>1,c)]\<bullet>[(x\<^isub>1 ,c)]\<bullet>e\<^isub>1 : T" using f f' 
-    by (auto simp add: perm_fresh_fresh)
-  then have "([(x\<^isub>1,c)]\<bullet>(c,Data \<sigma>\<^isub>1)) #\<Gamma> \<turnstile> e\<^isub>1 : T" by perm_simp
-  then have g1:"(x\<^isub>1, Data \<sigma>\<^isub>1)#\<Gamma> \<turnstile> e\<^isub>1 : T"  using f' by (auto simp add: fresh_atm calc_atm fresh_prod)
-    (* The second part of the proof is the same *)
-  obtain c::name where f':"c \<sharp> (x\<^isub>2,x\<^isub>2',e\<^isub>2,e\<^isub>2',\<Gamma>)" by (erule exists_fresh[OF fs_name1])
-  have e2':"[(x\<^isub>2,c)]\<bullet>e\<^isub>2 = [(x\<^isub>2',c)]\<bullet>e\<^isub>2'" using e2 f' by (auto simp add: alpha_fresh fresh_prod fresh_atm)
-  have "[(x\<^isub>2',c)]\<bullet>((x\<^isub>2',Data \<sigma>\<^isub>2)# \<Gamma>) \<turnstile> [(x\<^isub>2',c)]\<bullet>e\<^isub>2' : T" using h2 typing_eqvt' by blast
-  then have x:"(c,Data \<sigma>\<^isub>2)#([(x\<^isub>2',c)]\<bullet>\<Gamma>) \<turnstile> [(x\<^isub>2',c)]\<bullet>e\<^isub>2': T" using f' 
-    by (auto simp add: fresh_atm calc_atm)
-  have "x\<^isub>2' \<sharp> \<Gamma>" using h2 typing_implies_valid by auto
-  then have "(c,Data \<sigma>\<^isub>2)#\<Gamma> \<turnstile> [(x\<^isub>2 ,c)]\<bullet>e\<^isub>2 : T" using f' x e2' by (auto simp add: perm_fresh_fresh)
-  then have "[(x\<^isub>2,c)]\<bullet>((c,Data \<sigma>\<^isub>2)#\<Gamma>) \<turnstile> [(x\<^isub>2,c)]\<bullet>[(x\<^isub>2 ,c)]\<bullet>e\<^isub>2 : T" using typing_eqvt' by blast 
-  then have "([(x\<^isub>2,c)]\<bullet>(c,Data \<sigma>\<^isub>2))#\<Gamma> \<turnstile> [(x\<^isub>2,c)]\<bullet>[(x\<^isub>2 ,c)]\<bullet>e\<^isub>2 : T" using f f' 
-    by (auto simp add: perm_fresh_fresh)
-  then have "([(x\<^isub>2,c)]\<bullet>(c,Data \<sigma>\<^isub>2)) #\<Gamma> \<turnstile> e\<^isub>2 : T" by perm_simp
-  then have g2:"(x\<^isub>2,Data \<sigma>\<^isub>2)#\<Gamma> \<turnstile> e\<^isub>2 : T"  using f' by (auto simp add: fresh_atm calc_atm fresh_prod)
-  show ?thesis using g1 g2 prems by auto 
-qed
+using a1 a2
+by (cases rule: typing.strong_cases [where x="x"])
+   (auto simp add: abs_fresh fresh_ty alpha trm.inject)
+
+abbreviation
+  "sub_context" :: "(name\<times>ty) list \<Rightarrow> (name\<times>ty) list \<Rightarrow> bool" ("_ \<subseteq> _" [55,55] 55)
+where
+  "\<Gamma>\<^isub>1 \<subseteq> \<Gamma>\<^isub>2 \<equiv> \<forall>x T. (x,T)\<in>set \<Gamma>\<^isub>1 \<longrightarrow> (x,T)\<in>set \<Gamma>\<^isub>2"
 
 lemma weakening: 
-  fixes \<Gamma>\<^isub>1 \<Gamma>\<^isub>2 :: "(name\<times>ty) list"
+  fixes \<Gamma>\<^isub>1 \<Gamma>\<^isub>2::"(name\<times>ty) list"
   assumes "\<Gamma>\<^isub>1 \<turnstile> e: T" and "valid \<Gamma>\<^isub>2" and "\<Gamma>\<^isub>1 \<subseteq> \<Gamma>\<^isub>2"
   shows "\<Gamma>\<^isub>2 \<turnstile> e: T"
   using assms
 proof(nominal_induct \<Gamma>\<^isub>1 e T avoiding: \<Gamma>\<^isub>2 rule: typing.strong_induct)
   case (t_Lam x \<Gamma>\<^isub>1 T\<^isub>1 t T\<^isub>2 \<Gamma>\<^isub>2)
+  have vc: "x\<sharp>\<Gamma>\<^isub>2" by fact
   have ih: "\<lbrakk>valid ((x,T\<^isub>1)#\<Gamma>\<^isub>2); (x,T\<^isub>1)#\<Gamma>\<^isub>1 \<subseteq> (x,T\<^isub>1)#\<Gamma>\<^isub>2\<rbrakk> \<Longrightarrow> (x,T\<^isub>1)#\<Gamma>\<^isub>2 \<turnstile> t : T\<^isub>2" by fact
-  have H1: "valid \<Gamma>\<^isub>2" by fact
-  have H2: "\<Gamma>\<^isub>1 \<subseteq> \<Gamma>\<^isub>2" by fact
-  have fs: "x\<sharp>\<Gamma>\<^isub>2" by fact
-  then have "valid ((x,T\<^isub>1)#\<Gamma>\<^isub>2)" using H1 by auto
-  moreover have "(x,T\<^isub>1)#\<Gamma>\<^isub>1 \<subseteq> (x,T\<^isub>1)#\<Gamma>\<^isub>2" using H2 by auto
+  have "valid \<Gamma>\<^isub>2" by fact
+  then have "valid ((x,T\<^isub>1)#\<Gamma>\<^isub>2)" using vc by auto
+  moreover
+  have "\<Gamma>\<^isub>1 \<subseteq> \<Gamma>\<^isub>2" by fact
+  then have "(x,T\<^isub>1)#\<Gamma>\<^isub>1 \<subseteq> (x,T\<^isub>1)#\<Gamma>\<^isub>2" by simp
   ultimately have "(x,T\<^isub>1)#\<Gamma>\<^isub>2 \<turnstile> t : T\<^isub>2" using ih by simp 
-  thus "\<Gamma>\<^isub>2 \<turnstile> Lam [x].t : T\<^isub>1\<rightarrow>T\<^isub>2" using fs by auto
-next
-  case (t_Case x\<^isub>1 \<Gamma>\<^isub>1 e e\<^isub>2 x\<^isub>2 e\<^isub>1 S\<^isub>1 S\<^isub>2 T \<Gamma>\<^isub>2)
-  then have ih\<^isub>1: "valid ((x\<^isub>1,Data S\<^isub>1)#\<Gamma>\<^isub>2) \<Longrightarrow> (x\<^isub>1,Data S\<^isub>1)#\<Gamma>\<^isub>2 \<turnstile> e\<^isub>1 : T" 
-       and  ih\<^isub>2: "valid ((x\<^isub>2,Data S\<^isub>2)#\<Gamma>\<^isub>2) \<Longrightarrow> (x\<^isub>2,Data S\<^isub>2)#\<Gamma>\<^isub>2 \<turnstile> e\<^isub>2 : T" 
-       and  ih\<^isub>3: "\<Gamma>\<^isub>2 \<turnstile> e : Data (DSum S\<^isub>1 S\<^isub>2)" by auto 
-  have fs\<^isub>1: "x\<^isub>1\<sharp>\<Gamma>\<^isub>2" "x\<^isub>1\<sharp>e" "x\<^isub>1\<sharp>e\<^isub>2" "x\<^isub>1\<sharp>x\<^isub>2" by fact+
-  have fs\<^isub>2: "x\<^isub>2\<sharp>\<Gamma>\<^isub>2" "x\<^isub>2\<sharp>e" "x\<^isub>2\<sharp>e\<^isub>1" "x\<^isub>2\<sharp>x\<^isub>1" by fact+
-  have "valid \<Gamma>\<^isub>2" by fact
-  then have "valid ((x\<^isub>1,Data S\<^isub>1)#\<Gamma>\<^isub>2)" and "valid ((x\<^isub>2,Data S\<^isub>2)#\<Gamma>\<^isub>2)" using fs\<^isub>1 fs\<^isub>2 by auto
-  then have "(x\<^isub>1, Data S\<^isub>1)#\<Gamma>\<^isub>2 \<turnstile> e\<^isub>1 : T" and "(x\<^isub>2, Data S\<^isub>2)#\<Gamma>\<^isub>2 \<turnstile> e\<^isub>2 : T" using ih\<^isub>1 ih\<^isub>2 by simp_all
-  with ih\<^isub>3 show "\<Gamma>\<^isub>2 \<turnstile> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 : T" using fs\<^isub>1 fs\<^isub>2 by auto
+  with vc show "\<Gamma>\<^isub>2 \<turnstile> Lam [x].t : T\<^isub>1\<rightarrow>T\<^isub>2" by auto
 qed (auto)
 
 lemma context_exchange:
   assumes a: "(x\<^isub>1,T\<^isub>1)#(x\<^isub>2,T\<^isub>2)#\<Gamma> \<turnstile> e : T"
   shows "(x\<^isub>2,T\<^isub>2)#(x\<^isub>1,T\<^isub>1)#\<Gamma> \<turnstile> e : T"
 proof -
   from  a have "valid ((x\<^isub>1,T\<^isub>1)#(x\<^isub>2,T\<^isub>2)#\<Gamma>)" by (simp add: typing_implies_valid)
   then have "x\<^isub>1\<noteq>x\<^isub>2" "x\<^isub>1\<sharp>\<Gamma>" "x\<^isub>2\<sharp>\<Gamma>" "valid \<Gamma>"
     by (auto simp: fresh_list_cons fresh_atm[symmetric])
   then have "valid ((x\<^isub>2,T\<^isub>2)#(x\<^isub>1,T\<^isub>1)#\<Gamma>)"
-    by (auto simp: fresh_list_cons fresh_atm)
+    by (auto simp: fresh_list_cons fresh_atm fresh_ty)
   moreover 
   have "(x\<^isub>1,T\<^isub>1)#(x\<^isub>2,T\<^isub>2)#\<Gamma> \<subseteq> (x\<^isub>2,T\<^isub>2)#(x\<^isub>1,T\<^isub>1)#\<Gamma>" by auto
   ultimately show "(x\<^isub>2,T\<^isub>2)#(x\<^isub>1,T\<^isub>1)#\<Gamma> \<turnstile> e : T" using a by (auto intro: weakening)
 qed
 
 lemma typing_var_unicity: 
-  assumes "(x,t\<^isub>1)#\<Gamma> \<turnstile> Var x : t\<^isub>2"
-  shows "t\<^isub>1=t\<^isub>2"
+  assumes a: "(x,T\<^isub>1)#\<Gamma> \<turnstile> Var x : T\<^isub>2"
+  shows "T\<^isub>1=T\<^isub>2"
 proof - 
-  have "(x,t\<^isub>2) \<in> set ((x,t\<^isub>1)#\<Gamma>)" and "valid ((x,t\<^isub>1)#\<Gamma>)" using assms by auto
-  thus "t\<^isub>1=t\<^isub>2" by (simp only: type_unicity_in_context)
+  have "(x,T\<^isub>2) \<in> set ((x,T\<^isub>1)#\<Gamma>)" using a
+    by (cases) (auto simp add: trm.inject)
+  moreover 
+  have "valid ((x,T\<^isub>1)#\<Gamma>)" using a by (simp add: typing_implies_valid)
+  ultimately show "T\<^isub>1=T\<^isub>2" by (simp only: type_unicity_in_context)
 qed
 
 lemma typing_substitution: 
   fixes \<Gamma>::"(name \<times> ty) list"
   assumes "(x,T')#\<Gamma> \<turnstile> e : T" 

     then have "T=T'" using h1 typing_var_unicity by auto
     then show "\<Gamma> \<turnstile> (Var y)[x::=e'] : T" using as h2 by simp
   next
     case False
     assume as: "x\<noteq>y" 
-    have "(y,T) \<in> set ((x,T')#\<Gamma>)" using h1 by auto
+    have "(y,T) \<in> set ((x,T')#\<Gamma>)" using h1 by (cases) (auto simp add: trm.inject)
     then have "(y,T) \<in> set \<Gamma>" using as by auto
     moreover 
     have "valid \<Gamma>" using h2 by (simp only: typing_implies_valid)
     ultimately show "\<Gamma> \<turnstile> (Var y)[x::=e'] : T" using as by (simp add: t_Var)
   qed
 next
   case (Lam y t \<Gamma> e' x T)
-  have vc: "y\<sharp>\<Gamma>" "y\<sharp>x" "y\<sharp>e'"  by fact+
+  have vc: "y\<sharp>\<Gamma>" "y\<sharp>x" "y\<sharp>e'" by fact
   have pr1: "\<Gamma> \<turnstile> e' : T'" by fact
   have pr2: "(x,T')#\<Gamma> \<turnstile> Lam [y].t : T" by fact
   then obtain T\<^isub>1 T\<^isub>2 where pr2': "(y,T\<^isub>1)#(x,T')#\<Gamma> \<turnstile> t : T\<^isub>2" and eq: "T = T\<^isub>1\<rightarrow>T\<^isub>2" 
-    using vc by (auto simp add: fresh_list_cons)
+    using vc by (auto elim: t_Lam_elim simp add: fresh_list_cons fresh_ty)
   then have pr2'':"(x,T')#(y,T\<^isub>1)#\<Gamma> \<turnstile> t : T\<^isub>2" by (simp add: context_exchange)
   have ih: "\<lbrakk>(x,T')#(y,T\<^isub>1)#\<Gamma> \<turnstile> t : T\<^isub>2; (y,T\<^isub>1)#\<Gamma> \<turnstile> e' : T'\<rbrakk> \<Longrightarrow> (y,T\<^isub>1)#\<Gamma> \<turnstile> t[x::=e'] : T\<^isub>2" by fact
   have "valid \<Gamma>" using pr1 by (simp add: typing_implies_valid)
   then have "valid ((y,T\<^isub>1)#\<Gamma>)" using vc by auto
-  then have "(y,T\<^isub>1)#\<Gamma> \<turnstile> e' : T'" using pr1 by (auto intro: weakening)
+  then have "(y,T\<^isub>1)#\<Gamma> \<turnstile> e' : T'" using pr1 by (simp add: weakening)
   then have "(y,T\<^isub>1)#\<Gamma> \<turnstile> t[x::=e'] : T\<^isub>2" using ih pr2'' by simp
-  then have "\<Gamma> \<turnstile> Lam [y].(t[x::=e']) : T\<^isub>1\<rightarrow>T\<^isub>2" using vc by (auto intro: t_Lam)
-  thus "\<Gamma> \<turnstile> (Lam [y].t)[x::=e'] : T" using vc eq by simp
+  then have "\<Gamma> \<turnstile> Lam [y].(t[x::=e']) : T\<^isub>1\<rightarrow>T\<^isub>2" using vc by auto
+  then show "\<Gamma> \<turnstile> (Lam [y].t)[x::=e'] : T" using vc eq by simp
 next
-  case (Case t\<^isub>1 x\<^isub>1 t\<^isub>2 x\<^isub>2 t3 \<Gamma> e' x T)
-  have vc: "x\<^isub>1\<sharp>\<Gamma>" "x\<^isub>1\<sharp>e'" "x\<^isub>1\<sharp>x""x\<^isub>1\<sharp>t\<^isub>1" "x\<^isub>1\<sharp>t3" "x\<^isub>2\<sharp>\<Gamma>" 
-           "x\<^isub>2\<sharp>e'" "x\<^isub>2\<sharp>x"  "x\<^isub>2\<sharp>t\<^isub>1" "x\<^isub>2\<sharp>t\<^isub>2" "x\<^isub>2\<noteq>x\<^isub>1" by fact+
-  have as1: "\<Gamma> \<turnstile> e' : T'" by fact
-  have as2: "(x,T')#\<Gamma> \<turnstile> Case t\<^isub>1 of inl x\<^isub>1 \<rightarrow> t\<^isub>2 | inr x\<^isub>2 \<rightarrow> t3 : T" by fact
-  then obtain S\<^isub>1 S\<^isub>2 where 
-    h1:"(x,T')#\<Gamma> \<turnstile> t\<^isub>1 : Data (DSum S\<^isub>1 S\<^isub>2)" and
-    h2:"(x\<^isub>1,Data S\<^isub>1)#(x,T')#\<Gamma> \<turnstile> t\<^isub>2 : T" and
-    h3:"(x\<^isub>2,Data S\<^isub>2)#(x,T')#\<Gamma> \<turnstile> t3 : T"
-    using vc by (auto simp add: fresh_list_cons)
-  have ih1: "\<lbrakk>(x,T')#\<Gamma> \<turnstile> t\<^isub>1 : Data (DSum S\<^isub>1 S\<^isub>2); \<Gamma> \<turnstile> e' : T'\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> t\<^isub>1[x::=e'] : Data (DSum S\<^isub>1 S\<^isub>2)"
-  and  ih2: "\<lbrakk>(x,T')#(x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>2:T; (x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> e':T'\<rbrakk> \<Longrightarrow> (x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>2[x::=e']:T" 
-  and  ih3: "\<lbrakk>(x,T')#(x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t3:T; (x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> e':T'\<rbrakk> \<Longrightarrow> (x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t3[x::=e']:T"
-    by fact+
-  from h2 have h2': "(x,T')#(x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>2 : T" by (rule context_exchange)
-  from h3 have h3': "(x,T')#(x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t3 : T" by (rule context_exchange)
-  have "\<Gamma> \<turnstile> t\<^isub>1[x::=e'] : Data (DSum S\<^isub>1 S\<^isub>2)" using h1 ih1 as1 by simp
-  moreover
-  have "valid ((x\<^isub>1,Data S\<^isub>1)#\<Gamma>)" using h2' by (auto dest: typing_implies_valid)
-  then have "(x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> e' : T'" using as1 by (auto simp add: weakening)
-  then have "(x\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>2[x::=e'] : T" using ih2 h2' by simp
-  moreover 
-  have "valid ((x\<^isub>2,Data S\<^isub>2)#\<Gamma>)" using h3' by (auto dest: typing_implies_valid)
-  then have "(x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> e' : T'" using as1 by (auto simp add: weakening)
-  then have "(x\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t3[x::=e'] : T" using ih3 h3' by simp
-  ultimately have "\<Gamma> \<turnstile> Case (t\<^isub>1[x::=e']) of inl x\<^isub>1 \<rightarrow> (t\<^isub>2[x::=e']) | inr x\<^isub>2 \<rightarrow> (t3[x::=e']) : T"
-    using vc by (auto simp add: fresh_atm fresh_subst)
-  thus "\<Gamma> \<turnstile> (Case t\<^isub>1 of inl x\<^isub>1 \<rightarrow> t\<^isub>2 | inr x\<^isub>2 \<rightarrow> t3)[x::=e'] : T" using vc by simp
-qed (simp, fast)+
+  case (App e1 e2 \<Gamma> e' x T)
+  have "(x,T')#\<Gamma> \<turnstile> App e1 e2 : T" by fact
+  then obtain Tn where a1: "(x,T')#\<Gamma> \<turnstile> e1 : Tn \<rightarrow> T"
+                   and a2: "(x,T')#\<Gamma> \<turnstile> e2 : Tn"
+    by (cases) (auto simp add: trm.inject)
+  have a3: "\<Gamma> \<turnstile> e' : T'" by fact
+  have ih1: "\<lbrakk>(x,T')#\<Gamma> \<turnstile> e1 : Tn\<rightarrow>T; \<Gamma> \<turnstile> e' : T'\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> e1[x::=e'] : Tn\<rightarrow>T" by fact
+  have ih2: "\<lbrakk>(x,T')#\<Gamma> \<turnstile> e2 : Tn; \<Gamma> \<turnstile> e' : T'\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> e2[x::=e'] : Tn" by fact
+  then show ?case using a1 a2 a3 ih1 ih2 by auto 
+qed
+
+text {* Values *}
+inductive
+  val :: "trm\<Rightarrow>bool" 
+where
+  v_Lam[intro]:   "val (Lam [x].e)"
+
+equivariance val 
+
+lemma not_val_App[simp]:
+  shows 
+  "\<not> val (App e\<^isub>1 e\<^isub>2)" 
+  "\<not> val (Var x)"
+  by (auto elim: val.cases)
 
 text {* Big-Step Evaluation *}
 
 inductive
   big :: "trm\<Rightarrow>trm\<Rightarrow>bool" ("_ \<Down> _" [80,80] 80) 
 where
   b_Lam[intro]:   "Lam [x].e \<Down> Lam [x].e"
 | b_App[intro]:   "\<lbrakk>x\<sharp>(e\<^isub>1,e\<^isub>2,e'); e\<^isub>1\<Down>Lam [x].e; e\<^isub>2\<Down>e\<^isub>2'; e[x::=e\<^isub>2']\<Down>e'\<rbrakk> \<Longrightarrow> App e\<^isub>1 e\<^isub>2 \<Down> e'"
-| b_Const[intro]: "Const n \<Down> Const n"
-| b_Pr[intro]:    "\<lbrakk>e\<^isub>1\<Down>e\<^isub>1'; e\<^isub>2\<Down>e\<^isub>2'\<rbrakk> \<Longrightarrow> Pr e\<^isub>1 e\<^isub>2 \<Down> Pr e\<^isub>1' e\<^isub>2'"
-| b_Fst[intro]:   "e\<Down>Pr e\<^isub>1 e\<^isub>2 \<Longrightarrow> Fst e\<Down>e\<^isub>1"
-| b_Snd[intro]:   "e\<Down>Pr e\<^isub>1 e\<^isub>2 \<Longrightarrow> Snd e\<Down>e\<^isub>2"
-| b_InL[intro]:   "e\<Down>e' \<Longrightarrow> InL e \<Down> InL e'"
-| b_InR[intro]:   "e\<Down>e' \<Longrightarrow> InR e \<Down> InR e'"
-| b_CaseL[intro]: "\<lbrakk>x\<^isub>1\<sharp>(e,e\<^isub>2,e'',x\<^isub>2); x\<^isub>2\<sharp>(e,e\<^isub>1,e'',x\<^isub>1) ; e\<Down>InL e'; e\<^isub>1[x\<^isub>1::=e']\<Down>e''\<rbrakk> 
-                   \<Longrightarrow> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''"
-| b_CaseR[intro]: "\<lbrakk>x\<^isub>1\<sharp>(e,e\<^isub>2,e'',x\<^isub>2); x\<^isub>2\<sharp>(e,e\<^isub>1,e'',x\<^isub>1) ; e\<Down>InR e'; e\<^isub>2[x\<^isub>2::=e']\<Down>e''\<rbrakk> 
-                   \<Longrightarrow> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''"
 
 equivariance big
 
 nominal_inductive big
-  by (simp_all add: abs_fresh fresh_prod fresh_atm)
-
-lemma big_eqvt':
-  fixes pi::"name prm"
-  assumes a: "(pi\<bullet>t) \<Down> (pi\<bullet>t')"
-  shows "t \<Down> t'"
-using a
-apply -
-apply(drule_tac pi="rev pi" in big.eqvt)
-apply(perm_simp)
-done
-
-lemma fresh_preserved:
-  fixes x::name
-  fixes t::trm
-  fixes t'::trm
-  assumes "e \<Down> e'" and "x\<sharp>e" 
+  by (simp_all add: abs_fresh)
+
+lemma big_preserves_fresh:
+  fixes x::"name"
+  assumes a: "e \<Down> e'" "x\<sharp>e" 
   shows "x\<sharp>e'"
-  using assms by (induct) (auto simp add:fresh_subst')
-
-declare trm.inject  [simp add]
-declare ty.inject  [simp add]
-declare data.inject [simp add]
-
-inductive_cases b_inv_auto[elim]: 
-  "App e\<^isub>1 e\<^isub>2 \<Down> t" 
-  "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> t"
-  "Lam[x].t \<Down> t"
-  "Const n \<Down> t"
-  "Fst e \<Down> t"
-  "Snd e \<Down> t"
-  "InL e \<Down> t"
-  "InR e \<Down> t"
-  "Pr e\<^isub>1 e\<^isub>2 \<Down> t"
-
-declare trm.inject  [simp del]
-declare ty.inject  [simp del]
-declare data.inject [simp del]
-
-lemma b_App_elim[elim]:
-  assumes "App e\<^isub>1 e\<^isub>2 \<Down> e'" and "x\<sharp>(e\<^isub>1,e\<^isub>2,e')"
+  using a by (induct) (auto simp add: abs_fresh fresh_subst)
+
+lemma b_App_elim:
+  assumes a: "App e\<^isub>1 e\<^isub>2 \<Down> e'" "x\<sharp>(e\<^isub>1,e\<^isub>2,e')"
   obtains f\<^isub>1 and f\<^isub>2 where "e\<^isub>1 \<Down> Lam [x]. f\<^isub>1" "e\<^isub>2 \<Down> f\<^isub>2" "f\<^isub>1[x::=f\<^isub>2] \<Down> e'"
-  using assms
-  apply -
-  apply(erule b_inv_auto)
-  apply(drule_tac pi="[(xa,x)]" in big.eqvt)
-  apply(drule_tac pi="[(xa,x)]" in big.eqvt)
-  apply(drule_tac pi="[(xa,x)]" in big.eqvt)
-  apply(perm_simp add: calc_atm eqvts)
-  done
-
-lemma  b_CaseL_elim[elim]: 
-  assumes "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''" 
-  and     "\<And> t. \<not>  e \<Down> InR t"
-  and     "x\<^isub>1\<sharp>e''" "x\<^isub>1\<sharp>e" "x\<^isub>2\<sharp>e''" "x\<^isub>1\<sharp>e"
-  obtains e' where "e \<Down> InL e'" and "e\<^isub>1[x\<^isub>1::=e'] \<Down> e''"
-  using assms 
-  apply -
-  apply(rule b_inv_auto(2))
-  apply(auto)
-  apply(simp add: alpha)
-  apply(auto)
-  apply(drule_tac x="[(x\<^isub>1,x\<^isub>1')]\<bullet>e'" in meta_spec)
-  apply(drule meta_mp)
-  apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
-  apply(perm_simp add: fresh_prod)
-  apply(drule meta_mp)
-  apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
-  apply(perm_simp add: eqvts calc_atm)
-  apply(assumption)
-  apply(drule_tac x="[(x\<^isub>1,x\<^isub>1')]\<bullet>e'" in meta_spec)
-  apply(drule meta_mp)
-  apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
-  apply(perm_simp add: fresh_prod)
-  apply(drule meta_mp)
-  apply(rule_tac pi="[(x\<^isub>1,x\<^isub>1')]" in big_eqvt')
-  apply(perm_simp add: eqvts calc_atm)
-  apply(assumption)
-done
-
-lemma b_CaseR_elim[elim]: 
-  assumes "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> e''" 
-  and     "\<And> t. \<not> e \<Down> InL t"
-  and     "x\<^isub>1\<sharp>e''" "x\<^isub>1\<sharp>e" "x\<^isub>2\<sharp>e''" "x\<^isub>2\<sharp>e"
-  obtains e' where "e \<Down> InR e'" and "e\<^isub>2[x\<^isub>2::=e'] \<Down> e''"
-  using assms 
-  apply -
-  apply(rule b_inv_auto(2))
-  apply(auto)
-  apply(simp add: alpha)
-  apply(auto)
-  apply(drule_tac x="[(x\<^isub>2,x\<^isub>2')]\<bullet>e'" in meta_spec)
-  apply(drule meta_mp)
-  apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
-  apply(perm_simp add: fresh_prod)
-  apply(drule meta_mp)
-  apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
-  apply(perm_simp add: eqvts calc_atm)
-  apply(assumption)
-  apply(drule_tac x="[(x\<^isub>2,x\<^isub>2')]\<bullet>e'" in meta_spec)
-  apply(drule meta_mp)
-  apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
-  apply(perm_simp add: fresh_prod)
-  apply(drule meta_mp)
-  apply(rule_tac pi="[(x\<^isub>2,x\<^isub>2')]" in big_eqvt')
-  apply(perm_simp add: eqvts calc_atm)
-  apply(assumption)
-done
-
-inductive
-  val :: "trm\<Rightarrow>bool" 
-where
-  v_Lam[intro]:   "val (Lam [x].e)"
-| v_Const[intro]: "val (Const n)"
-| v_Pr[intro]:    "\<lbrakk>val e\<^isub>1; val e\<^isub>2\<rbrakk> \<Longrightarrow> val (Pr e\<^isub>1 e\<^isub>2)"
-| v_InL[intro]:   "val e \<Longrightarrow> val (InL e)"
-| v_InR[intro]:   "val e \<Longrightarrow> val (InR e)"
-
-declare trm.inject  [simp add]
-declare ty.inject  [simp add]
-declare data.inject [simp add]
-
-inductive_cases v_inv_auto[elim]: 
-  "val (Const n)"
-  "val (Pr e\<^isub>1 e\<^isub>2)"
-  "val (InL e)"
-  "val (InR e)"
-  "val (Fst e)"
-  "val (Snd e)"
-  "val (Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2)"
-  "val (Var x)"
-  "val (Lam [x].e)"
-  "val (App e\<^isub>1 e\<^isub>2)"
-
-declare trm.inject  [simp del]
-declare ty.inject  [simp del]
-declare data.inject [simp del]
+  using a
+by (cases rule: big.strong_cases[where x="x" and xa="x"])
+   (auto simp add: trm.inject)
 
 lemma subject_reduction:
-  assumes a: "e \<Down> e'" 
-  and     b: "\<Gamma> \<turnstile> e : T"
+  assumes a: "e \<Down> e'" and b: "\<Gamma> \<turnstile> e : T"
   shows "\<Gamma> \<turnstile> e' : T"
   using a b
 proof (nominal_induct avoiding: \<Gamma> arbitrary: T rule: big.strong_induct) 
   case (b_App x e\<^isub>1 e\<^isub>2 e' e e\<^isub>2' \<Gamma> T)
   have vc: "x\<sharp>\<Gamma>" by fact
   have "\<Gamma> \<turnstile> App e\<^isub>1 e\<^isub>2 : T" by fact
-  then obtain T' where 
-    a1: "\<Gamma> \<turnstile> e\<^isub>1 : T'\<rightarrow>T" and  
-    a2: "\<Gamma> \<turnstile> e\<^isub>2 : T'" by auto
+  then obtain T' where a1: "\<Gamma> \<turnstile> e\<^isub>1 : T'\<rightarrow>T" and a2: "\<Gamma> \<turnstile> e\<^isub>2 : T'" 
+    by (cases) (auto simp add: trm.inject)
   have ih1: "\<Gamma> \<turnstile> e\<^isub>1 : T' \<rightarrow> T \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].e : T' \<rightarrow> T" by fact
   have ih2: "\<Gamma> \<turnstile> e\<^isub>2 : T' \<Longrightarrow> \<Gamma> \<turnstile> e\<^isub>2' : T'" by fact 
   have ih3: "\<Gamma> \<turnstile> e[x::=e\<^isub>2'] : T \<Longrightarrow> \<Gamma> \<turnstile> e' : T" by fact
   have "\<Gamma> \<turnstile> Lam [x].e : T'\<rightarrow>T" using ih1 a1 by simp 
-  then have "((x,T')#\<Gamma>) \<turnstile> e : T" using vc by (auto simp add: ty.inject)
+  then have "((x,T')#\<Gamma>) \<turnstile> e : T" using vc  
+    by (auto elim: t_Lam_elim simp add: ty.inject)
   moreover
   have "\<Gamma> \<turnstile> e\<^isub>2': T'" using ih2 a2 by simp
   ultimately have "\<Gamma> \<turnstile> e[x::=e\<^isub>2'] : T" by (simp add: typing_substitution)
   thus "\<Gamma> \<turnstile> e' : T" using ih3 by simp
-next
-  case (b_CaseL x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' \<Gamma>)
-  have vc: "x\<^isub>1\<sharp>\<Gamma>" "x\<^isub>2\<sharp>\<Gamma>" by fact+
-  have "\<Gamma> \<turnstile> Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 : T" by fact 
-  then obtain S\<^isub>1 S\<^isub>2 e\<^isub>1' e\<^isub>2' where 
-    a1: "\<Gamma> \<turnstile> e : Data (DSum S\<^isub>1 S\<^isub>2)" and 
-    a2: "((x\<^isub>1,Data S\<^isub>1)#\<Gamma>) \<turnstile> e\<^isub>1 : T" using vc by auto 
-  have ih1:"\<Gamma> \<turnstile> e : Data (DSum S\<^isub>1 S\<^isub>2) \<Longrightarrow> \<Gamma> \<turnstile> InL e' : Data (DSum S\<^isub>1 S\<^isub>2)" by fact 
-  have ih2:"\<Gamma> \<turnstile> e\<^isub>1[x\<^isub>1::=e'] : T \<Longrightarrow> \<Gamma> \<turnstile> e'' : T " by fact 
-  have "\<Gamma> \<turnstile> InL e' : Data (DSum S\<^isub>1 S\<^isub>2)" using ih1 a1 by simp
-  then have "\<Gamma> \<turnstile> e' : Data S\<^isub>1" by auto
-  then have "\<Gamma> \<turnstile> e\<^isub>1[x\<^isub>1::=e'] : T" using a2 by (simp add: typing_substitution)
-  then show "\<Gamma> \<turnstile> e'' : T" using ih2 by simp
-next
- case (b_CaseR x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' \<Gamma> T)
- then show "\<Gamma> \<turnstile> e'' : T" by (blast intro: typing_substitution)
 qed (blast)+
 
 lemma unicity_of_evaluation:
   assumes a: "e \<Down> e\<^isub>1" 
   and     b: "e \<Down> e\<^isub>2"

   case (b_App x e\<^isub>1 e\<^isub>2 e' e\<^isub>1' e\<^isub>2' t\<^isub>2)
   have ih1: "\<And>t. e\<^isub>1 \<Down> t \<Longrightarrow> Lam [x].e\<^isub>1' = t" by fact
   have ih2:"\<And>t. e\<^isub>2 \<Down> t \<Longrightarrow> e\<^isub>2' = t" by fact
   have ih3: "\<And>t. e\<^isub>1'[x::=e\<^isub>2'] \<Down> t \<Longrightarrow> e' = t" by fact
   have app: "App e\<^isub>1 e\<^isub>2 \<Down> t\<^isub>2" by fact
-  have vc: "x\<sharp>e\<^isub>1" "x\<sharp>e\<^isub>2" by fact+
-  then have "x \<sharp> App e\<^isub>1 e\<^isub>2" by auto
-  then have vc': "x\<sharp>t\<^isub>2" using fresh_preserved app by blast
-  from vc vc' obtain f\<^isub>1 f\<^isub>2 where x1: "e\<^isub>1 \<Down> Lam [x]. f\<^isub>1" and x2: "e\<^isub>2 \<Down> f\<^isub>2" and x3: "f\<^isub>1[x::=f\<^isub>2] \<Down> t\<^isub>2"
-    using app by (auto simp add: fresh_prod)
+  have vc: "x\<sharp>e\<^isub>1" "x\<sharp>e\<^isub>2" "x\<sharp>t\<^isub>2" by fact
+  then have "x\<sharp>App e\<^isub>1 e\<^isub>2" by auto
+  from app vc obtain f\<^isub>1 f\<^isub>2 where x1: "e\<^isub>1 \<Down> Lam [x]. f\<^isub>1" and x2: "e\<^isub>2 \<Down> f\<^isub>2" and x3: "f\<^isub>1[x::=f\<^isub>2] \<Down> t\<^isub>2"
+    by (cases rule: big.strong_cases[where x="x" and xa="x"])
+       (auto simp add: trm.inject)
   then have "Lam [x]. f\<^isub>1 = Lam [x]. e\<^isub>1'" using ih1 by simp
   then 
   have "f\<^isub>1 = e\<^isub>1'" by (auto simp add: trm.inject alpha) 
   moreover 
   have "f\<^isub>2 = e\<^isub>2'" using x2 ih2 by simp
   ultimately have "e\<^isub>1'[x::=e\<^isub>2'] \<Down> t\<^isub>2" using x3 by simp
-  thus ?case using ih3 by simp
-next
-  case (b_CaseL  x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' t\<^isub>2)
-  have fs: "x\<^isub>1\<sharp>e" "x\<^isub>1\<sharp>t\<^isub>2" "x\<^isub>2\<sharp>e" "x\<^isub>2\<sharp>t\<^isub>2" by fact+
-  have ih1:"\<And>t. e \<Down> t \<Longrightarrow> InL e' = t" by fact 
-  have ih2:"\<And>t. e\<^isub>1[x\<^isub>1::=e'] \<Down> t \<Longrightarrow> e'' = t" by fact
-  have ha: "\<not>(\<exists>t. e \<Down> InR t)" using ih1 by force
-  have "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> t\<^isub>2" by fact
-  then obtain f' where "e \<Down> InL f'" and h: "e\<^isub>1[x\<^isub>1::=f']\<Down>t\<^isub>2" using ha fs by auto
-  then have "InL f' = InL e'" using ih1 by simp
-  then have "f' = e'" by (simp add: trm.inject)
-  then have "e\<^isub>1[x\<^isub>1::=e'] \<Down> t\<^isub>2" using h by simp
-  then show "e'' = t\<^isub>2" using ih2 by simp
-next 
-  case (b_CaseR x\<^isub>1 e e\<^isub>2 e'' x\<^isub>2 e\<^isub>1 e' t\<^isub>2 )
-  have fs: "x\<^isub>1\<sharp>e" "x\<^isub>1\<sharp>t\<^isub>2" "x\<^isub>2\<sharp>e" "x\<^isub>2\<sharp>t\<^isub>2" by fact+
-  have ih1: "\<And>t. e \<Down> t \<Longrightarrow> InR e' = t" by fact
-  have ih2: "\<And>t. e\<^isub>2[x\<^isub>2::=e'] \<Down> t \<Longrightarrow> e'' = t" by fact
-  have ha: "\<not>(\<exists>t. e \<Down> InL t)" using ih1 by force
-  have "Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2 \<Down> t\<^isub>2" by fact
-  then obtain f' where "e \<Down> InR f'" and h: "e\<^isub>2[x\<^isub>2::=f']\<Down>t\<^isub>2"  using ha fs by auto
-  then have "InR f' = InR e'" using ih1 by simp
-  then have "e\<^isub>2[x\<^isub>2::=e'] \<Down> t\<^isub>2" using h by (simp add: trm.inject)
-  thus "e'' = t\<^isub>2" using ih2 by simp
-next
-  case (b_Fst e e\<^isub>1 e\<^isub>2 e\<^isub>2')
-  have "e \<Down> Pr e\<^isub>1 e\<^isub>2" by fact
-  have "\<And> b. e \<Down> b \<Longrightarrow> Pr e\<^isub>1 e\<^isub>2 = b" by fact
-  have "Fst e \<Down> e\<^isub>2'" by fact
-  show "e\<^isub>1 = e\<^isub>2'" using prems by (force simp add: trm.inject)
-next
-  case (b_Snd e e\<^isub>1 e\<^isub>2 e\<^isub>2')
-  have "e \<Down> Pr e\<^isub>1 e\<^isub>2" by fact
-  have "\<And> b. e \<Down> b \<Longrightarrow> Pr e\<^isub>1 e\<^isub>2 = b" by fact
-  have "Snd e \<Down> e\<^isub>2'" by fact
-  show "e\<^isub>2 = e\<^isub>2'" using prems by (force simp add: trm.inject)
-qed (blast)+
-
-lemma not_val_App[simp]:
-  shows 
-  "\<not> val (App e\<^isub>1 e\<^isub>2)" 
-  "\<not> val (Fst e)"
-  "\<not> val (Snd e)"
-  "\<not> val (Var x)"
-  "\<not> val (Case e of inl x\<^isub>1 \<rightarrow> e\<^isub>1 | inr x\<^isub>2 \<rightarrow> e\<^isub>2)"
-by auto
+  thus "e' = t\<^isub>2" using ih3 by simp
+qed
 
 lemma reduces_evaluates_to_values:
   assumes h:"t \<Down> t'"
   shows "val t'"
   using h by (induct) (auto)
-
-lemma type_prod_evaluates_to_pairs:
-  assumes a: "\<Gamma> \<turnstile> t : Data (DProd S\<^isub>1 S\<^isub>2)" 
-  and     b: "t \<Down> t'"
-  obtains t\<^isub>1 t\<^isub>2 where "t' = Pr t\<^isub>1 t\<^isub>2"
-proof -
-   have "\<Gamma> \<turnstile> t' : Data (DProd S\<^isub>1 S\<^isub>2)" using assms subject_reduction by simp
-   moreover
-   have "val t'" using reduces_evaluates_to_values assms by simp
-   ultimately obtain t\<^isub>1 t\<^isub>2 where "t' = Pr t\<^isub>1 t\<^isub>2" by (cases, auto simp add:ty.inject data.inject)
-   thus ?thesis using prems by auto
-qed
-
-lemma type_sum_evaluates_to_ins:
-  assumes "\<Gamma> \<turnstile> t : Data (DSum \<sigma>\<^isub>1 \<sigma>\<^isub>2)" and "t \<Down> t'"
-  shows "(\<exists>t''. t' = InL t'') \<or> (\<exists>t''. t' = InR t'')"
-proof -
-  have "\<Gamma> \<turnstile> t' : Data (DSum \<sigma>\<^isub>1 \<sigma>\<^isub>2)" using assms subject_reduction by simp
-  moreover
-  have "val t'" using reduces_evaluates_to_values assms by simp
-  ultimately obtain t'' where "t' = InL t'' \<or>  t' = InR t''"
-    by (cases, auto simp add:ty.inject data.inject)
-  thus ?thesis by auto
-qed
 
 lemma type_arrow_evaluates_to_lams:
   assumes "\<Gamma> \<turnstile> t : \<sigma> \<rightarrow> \<tau>" and "t \<Down> t'"
   obtains  x t'' where "t' = Lam [x]. t''"
 proof -
   have "\<Gamma> \<turnstile> t' : \<sigma> \<rightarrow> \<tau>" using assms subject_reduction by simp
   moreover
   have "val t'" using reduces_evaluates_to_values assms by simp
-  ultimately obtain x t'' where "t' = Lam [x]. t''" by (cases, auto simp add:ty.inject data.inject)
+  ultimately obtain x t'' where "t' = Lam [x]. t''" by (cases, auto simp add:ty.inject)
   thus ?thesis using prems by auto
 qed
 
-lemma type_nat_evaluates_to_consts:
-  assumes "\<Gamma> \<turnstile> t : Data DNat" and "t \<Down> t'"
-  obtains n where "t' = Const n"
-proof -
-  have "\<Gamma> \<turnstile> t' : Data DNat " using assms subject_reduction by simp
-  moreover have "val t'" using reduces_evaluates_to_values assms by simp
-  ultimately obtain n where "t' = Const n" by (cases, auto simp add:ty.inject data.inject)
-  thus ?thesis using prems by auto
-qed
-
-consts
-  V' :: "data \<Rightarrow> trm set" 
-
-nominal_primrec    
-  "V' (DNat) = {Const n | n. n \<in> (UNIV::nat set)}"
-  "V' (DProd S\<^isub>1 S\<^isub>2) = {Pr x y | x y. x \<in> V' S\<^isub>1 \<and> y \<in> V' S\<^isub>2}"
-  "V' (DSum S\<^isub>1 S\<^isub>2) = {InL x | x. x \<in> V' S\<^isub>1} \<union> {InR y | y. y \<in> V' S\<^isub>2}"
-apply(rule TrueI)+
-done
-
-lemma Vprimes_are_values :
-  fixes S::"data"
-  assumes h: "e \<in> V' S"
-  shows "val e"
-using h 
-by (nominal_induct S arbitrary: e rule:data.induct)
-   (auto)
-
-lemma V'_eqvt:
-  fixes pi::"name prm"
-  assumes a: "v \<in> V' S"
-  shows "(pi\<bullet>v) \<in> V' S"
-using a
-by (nominal_induct S arbitrary: v rule: data.induct)
-   (auto simp add: trm.inject)
-
+(* Valuation *)
 consts
   V :: "ty \<Rightarrow> trm set" 
 
 nominal_primrec
-  "V (Data S) = V' S"
+  "V (TVar x) = {e. val e}"
   "V (T\<^isub>1 \<rightarrow> T\<^isub>2) = {Lam [x].e | x e. \<forall> v \<in> (V T\<^isub>1). \<exists> v'. e[x::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2}"
-apply(rule TrueI)+ 
-done
-
+  by (rule TrueI)+ 
+
+(* can go with strong inversion rules *)
 lemma V_eqvt:
   fixes pi::"name prm"
   assumes a: "x\<in>V T"
   shows "(pi\<bullet>x)\<in>V T"
 using a
 apply(nominal_induct T arbitrary: pi x rule: ty.induct)
 apply(auto simp add: trm.inject perm_set_def)
-apply(perm_simp add: V'_eqvt)
+apply(simp add: eqvts)
 apply(rule_tac x="pi\<bullet>xa" in exI)
 apply(rule_tac x="pi\<bullet>e" in exI)
 apply(simp)
 apply(auto)
 apply(drule_tac x="(rev pi)\<bullet>v" in bspec)

 apply(auto)
 apply(drule_tac pi="pi" in big.eqvt)
 apply(perm_simp add: eqvts)
 done
 
-lemma V_arrow_elim_weak[elim] :
+lemma V_arrow_elim_weak:
   assumes h:"u \<in> (V (T\<^isub>1 \<rightarrow> T\<^isub>2))"
   obtains a t where "u = Lam[a].t" and "\<forall> v \<in> (V T\<^isub>1). \<exists> v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2"
 using h by (auto)
 
-lemma V_arrow_elim_strong[elim]:
+lemma V_arrow_elim_strong:
   fixes c::"'a::fs_name"
   assumes h: "u \<in> V (T\<^isub>1 \<rightarrow> T\<^isub>2)"
   obtains a t where "a\<sharp>c" "u = Lam[a].t" "\<forall>v \<in> (V T\<^isub>1). \<exists> v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2"
 using h
 apply -

 (*A*)
 apply(rule exists_fresh')
 apply(simp add: fin_supp)
 done
 
-lemma V_are_values :
-  fixes T::"ty"
-  assumes h:"e \<in> V T"
+lemma Vs_are_values:
+  assumes a: "e \<in> V T"
   shows "val e"
-using h by (nominal_induct T arbitrary: e rule:ty.induct, auto simp add: Vprimes_are_values)
+using a by (nominal_induct T arbitrary: e rule: ty.induct) (auto)
 
 lemma values_reduce_to_themselves:
-  assumes h:"val v"
+  assumes a: "val v"
   shows "v \<Down> v"
-using h by (induct,auto)
-
-lemma Vs_reduce_to_themselves[simp]:
-  assumes h:"v \<in> V T"
+using a by (induct) (auto)
+
+lemma Vs_reduce_to_themselves:
+  assumes a: "v \<in> V T"
   shows "v \<Down> v"
-using h by (simp add: values_reduce_to_themselves V_are_values)
-
-lemma V_sum:
-  assumes h:"x \<in> V (Data (DSum S\<^isub>1 S\<^isub>2))"
-  shows "(\<exists> y. x= InL y \<and>  y \<in> V' S\<^isub>1) \<or> (\<exists> y. x= InR y \<and> y \<in> V' S\<^isub>2)"
-using h by simp
-
+using a by (simp add: values_reduce_to_themselves Vs_are_values)
+
+text {* '\<theta> maps x to e' asserts that \<theta> substitutes x with e *}
 abbreviation 
- mapsto :: "(name\<times>trm) list \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> bool" ("_ maps _ to _" [55,55,55] 55) 
-where
- "\<theta> maps x to e\<equiv> (lookup \<theta> x) = e"
+  mapsto :: "(name\<times>trm) list \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> bool" ("_ maps _ to _" [55,55,55] 55) 
+where
+ "\<theta> maps x to e \<equiv> (lookup \<theta> x) = e"
 
 abbreviation 
   v_closes :: "(name\<times>trm) list \<Rightarrow> (name\<times>ty) list \<Rightarrow> bool" ("_ Vcloses _" [55,55] 55) 
 where
-  "\<theta> Vcloses \<Gamma> \<equiv> \<forall>x T. ((x,T) \<in> set \<Gamma> \<longrightarrow> (\<exists>v. \<theta> maps x to v \<and> v \<in> (V T)))"
+  "\<theta> Vcloses \<Gamma> \<equiv> \<forall>x T. (x,T) \<in> set \<Gamma> \<longrightarrow> (\<exists>v. \<theta> maps x to v \<and> v \<in> V T)"
 
 lemma monotonicity:
   fixes m::"name"
   fixes \<theta>::"(name \<times> trm) list" 
   assumes h1: "\<theta> Vcloses \<Gamma>"

   }
   ultimately show "\<exists>e'.  ((x,e)#\<theta>) maps x' to e'  \<and> e' \<in> V T'" by blast
 qed
 
 lemma termination_aux:
-  fixes T :: "ty"
-  fixes \<Gamma> :: "(name \<times> ty) list"
-  fixes \<theta> :: "(name \<times> trm) list"
-  fixes e :: "trm"
   assumes h1: "\<Gamma> \<turnstile> e : T"
   and     h2: "\<theta> Vcloses \<Gamma>"
   shows "\<exists>v. \<theta><e> \<Down> v \<and> v \<in> V T" 
 using h2 h1
 proof(nominal_induct e avoiding: \<Gamma> \<theta> arbitrary: T rule: trm.induct)
   case (App e\<^isub>1 e\<^isub>2 \<Gamma> \<theta> T)
   have ih\<^isub>1:"\<And>\<theta> \<Gamma> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> e\<^isub>1 : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><e\<^isub>1> \<Down> v \<and> v \<in> V T" by fact
   have ih\<^isub>2:"\<And>\<theta> \<Gamma> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> e\<^isub>2 : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><e\<^isub>2> \<Down> v \<and> v \<in> V T" by fact
   have as\<^isub>1:"\<theta> Vcloses \<Gamma>" by fact 
   have as\<^isub>2: "\<Gamma> \<turnstile> App e\<^isub>1 e\<^isub>2 : T" by fact
-  from as\<^isub>2 obtain T' where "\<Gamma> \<turnstile> e\<^isub>1 : T' \<rightarrow> T" and "\<Gamma> \<turnstile> e\<^isub>2 : T'" by auto
+  then obtain T' where "\<Gamma> \<turnstile> e\<^isub>1 : T' \<rightarrow> T" and "\<Gamma> \<turnstile> e\<^isub>2 : T'" 
+    by (cases) (auto simp add: trm.inject)
   then obtain v\<^isub>1 v\<^isub>2 where "(i)": "\<theta><e\<^isub>1> \<Down> v\<^isub>1" "v\<^isub>1 \<in> V (T' \<rightarrow> T)"
                       and "(ii)":"\<theta><e\<^isub>2> \<Down> v\<^isub>2" "v\<^isub>2 \<in> V T'" using ih\<^isub>1 ih\<^isub>2 as\<^isub>1 by blast
   from "(i)" obtain x e' 
             where "v\<^isub>1 = Lam[x].e'" 
             and "(iii)": "(\<forall>v \<in> (V T').\<exists> v'. e'[x::=v] \<Down> v' \<and> v' \<in> V T)"
             and "(iv)":  "\<theta><e\<^isub>1> \<Down> Lam [x].e'" 
-            and fr: "x\<sharp>(\<theta>,e\<^isub>1,e\<^isub>2)" by blast
+            and fr: "x\<sharp>(\<theta>,e\<^isub>1,e\<^isub>2)" by (blast elim: V_arrow_elim_strong)
   from fr have fr\<^isub>1: "x\<sharp>\<theta><e\<^isub>1>" and fr\<^isub>2: "x\<sharp>\<theta><e\<^isub>2>" by (simp_all add: fresh_psubst)
   from "(ii)" "(iii)" obtain v\<^isub>3 where "(v)": "e'[x::=v\<^isub>2] \<Down> v\<^isub>3" "v\<^isub>3 \<in> V T" by auto
-  from fr\<^isub>2 "(ii)" have "x\<sharp>v\<^isub>2" by (simp add: fresh_preserved)
-  then have "x\<sharp>e'[x::=v\<^isub>2]" by (simp add: fresh_subst_fresh)
-  then have fr\<^isub>3: "x\<sharp>v\<^isub>3" using "(v)" by (simp add: fresh_preserved)
+  from fr\<^isub>2 "(ii)" have "x\<sharp>v\<^isub>2" by (simp add: big_preserves_fresh)
+  then have "x\<sharp>e'[x::=v\<^isub>2]" by (simp add: fresh_subst')
+  then have fr\<^isub>3: "x\<sharp>v\<^isub>3" using "(v)" by (simp add: big_preserves_fresh)
   from fr\<^isub>1 fr\<^isub>2 fr\<^isub>3 have "x\<sharp>(\<theta><e\<^isub>1>,\<theta><e\<^isub>2>,v\<^isub>3)" by simp
   with "(iv)" "(ii)" "(v)" have "App (\<theta><e\<^isub>1>) (\<theta><e\<^isub>2>) \<Down> v\<^isub>3" by auto
   then show "\<exists>v. \<theta><App e\<^isub>1 e\<^isub>2> \<Down> v \<and> v \<in> V T" using "(v)" by auto
 next
-  case (Pr t\<^isub>1 t\<^isub>2 \<Gamma> \<theta> T)
-  have "\<Gamma> \<turnstile> Pr t\<^isub>1 t\<^isub>2 : T" by fact
-  then obtain T\<^isub>a T\<^isub>b where ta:"\<Gamma> \<turnstile> t\<^isub>1 : Data T\<^isub>a" and "\<Gamma> \<turnstile> t\<^isub>2 : Data T\<^isub>b" 
-                      and eq:"T=Data (DProd T\<^isub>a T\<^isub>b)" by auto
-  have h:"\<theta> Vcloses \<Gamma>" by fact
-  then obtain v\<^isub>1 v\<^isub>2 where "\<theta><t\<^isub>1> \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V (Data T\<^isub>a)" "\<theta><t\<^isub>2> \<Down> v\<^isub>2 \<and> v\<^isub>2 \<in> V (Data T\<^isub>b)" 
-    using prems by blast
-  thus "\<exists>v. \<theta><Pr t\<^isub>1 t\<^isub>2> \<Down> v \<and> v \<in> V T" using eq by auto
-next 
   case (Lam x e \<Gamma> \<theta> T)
   have ih:"\<And>\<theta> \<Gamma> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> e : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><e> \<Down> v \<and> v \<in> V T" by fact
   have as\<^isub>1: "\<theta> Vcloses \<Gamma>" by fact
   have as\<^isub>2: "\<Gamma> \<turnstile> Lam [x].e : T" by fact
-  have fs: "x\<sharp>\<Gamma>" "x\<sharp>\<theta>" by fact+
+  have fs: "x\<sharp>\<Gamma>" "x\<sharp>\<theta>" by fact
   from as\<^isub>2 fs obtain T\<^isub>1 T\<^isub>2 
-    where "(i)": "(x,T\<^isub>1)#\<Gamma> \<turnstile> e:T\<^isub>2" and "(ii)": "T = T\<^isub>1 \<rightarrow> T\<^isub>2" by auto
+    where "(i)": "(x,T\<^isub>1)#\<Gamma> \<turnstile> e:T\<^isub>2" and "(ii)": "T = T\<^isub>1 \<rightarrow> T\<^isub>2" using fs
+    by (cases rule: typing.strong_cases[where x="x"])
+       (auto simp add: trm.inject alpha abs_fresh fresh_ty)
   from "(i)" have "(iii)": "valid ((x,T\<^isub>1)#\<Gamma>)" by (simp add: typing_implies_valid)
   have "\<forall>v \<in> (V T\<^isub>1). \<exists>v'. (\<theta><e>)[x::=v] \<Down> v' \<and> v' \<in> V T\<^isub>2"
   proof
     fix v
     assume "v \<in> (V T\<^isub>1)"

   qed
   then have "Lam[x].\<theta><e> \<in> V (T\<^isub>1 \<rightarrow> T\<^isub>2)" by auto
   then have "\<theta><Lam [x].e> \<Down> Lam[x].\<theta><e> \<and> Lam[x].\<theta><e> \<in> V (T\<^isub>1\<rightarrow>T\<^isub>2)" using fs by auto
   thus "\<exists>v. \<theta><Lam [x].e> \<Down> v \<and> v \<in> V T" using "(ii)" by auto
 next
-  case (Case t' n\<^isub>1 t\<^isub>1 n\<^isub>2 t\<^isub>2 \<Gamma> \<theta> T)
-  have f: "n\<^isub>1\<sharp>\<Gamma>" "n\<^isub>1\<sharp>\<theta>" "n\<^isub>2\<sharp>\<Gamma>" "n\<^isub>2\<sharp>\<theta>" "n\<^isub>2\<noteq>n\<^isub>1" "n\<^isub>1\<sharp>t'"
-  "n\<^isub>1\<sharp>t\<^isub>2" "n\<^isub>2\<sharp>t'" "n\<^isub>2\<sharp>t\<^isub>1" by fact+
-  have h:"\<theta> Vcloses \<Gamma>" by fact
-  have th:"\<Gamma> \<turnstile> Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2 : T" by fact
-  then obtain S\<^isub>1 S\<^isub>2 where 
-    hm:"\<Gamma> \<turnstile> t' : Data (DSum S\<^isub>1 S\<^isub>2)" and
-    hl:"(n\<^isub>1,Data S\<^isub>1)#\<Gamma> \<turnstile> t\<^isub>1 : T" and
-    hr:"(n\<^isub>2,Data S\<^isub>2)#\<Gamma> \<turnstile> t\<^isub>2 : T" using f by auto
-  then obtain v\<^isub>0 where ht':"\<theta><t'> \<Down> v\<^isub>0" and hS:"v\<^isub>0 \<in> V (Data (DSum S\<^isub>1 S\<^isub>2))" using prems h by blast
-  (* We distinguish between the cases InL and InR *)
-  { fix v\<^isub>0'
-    assume eqc:"v\<^isub>0 = InL v\<^isub>0'" and "v\<^isub>0' \<in> V' S\<^isub>1"
-    then have inc: "v\<^isub>0' \<in> V (Data S\<^isub>1)" by auto
-    have "valid \<Gamma>" using th typing_implies_valid by auto
-    then moreover have "valid ((n\<^isub>1,Data S\<^isub>1)#\<Gamma>)" using f by auto
-    then moreover have "(n\<^isub>1,v\<^isub>0')#\<theta> Vcloses (n\<^isub>1,Data S\<^isub>1)#\<Gamma>" 
-      using inc h monotonicity by blast
-    moreover 
-    have ih:"\<And>\<Gamma> \<theta> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> t\<^isub>1 : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><t\<^isub>1> \<Down> v \<and> v \<in> V T" by fact
-    ultimately obtain v\<^isub>1 where ho: "((n\<^isub>1,v\<^isub>0')#\<theta>)<t\<^isub>1> \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V T" using hl by blast
-    then have r:"\<theta><t\<^isub>1>[n\<^isub>1::=v\<^isub>0'] \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V T" using psubst_subst_psubst f by simp
-    then moreover have "n\<^isub>1\<sharp>(\<theta><t'>,\<theta><t\<^isub>2>,v\<^isub>1,n\<^isub>2)" 
-      proof -
-	have "n\<^isub>1\<sharp>v\<^isub>0" using ht' fresh_preserved fresh_psubst f by auto 
-	then have "n\<^isub>1\<sharp>v\<^isub>0'" using eqc by auto
-	then have "n\<^isub>1\<sharp>v\<^isub>1" using f r fresh_preserved fresh_subst_fresh by blast
-	thus "n\<^isub>1\<sharp>(\<theta><t'>,\<theta><t\<^isub>2>,v\<^isub>1,n\<^isub>2)" using f by (simp add: fresh_atm fresh_psubst)
-      qed
-    moreover have "n\<^isub>2\<sharp>(\<theta><t'>,\<theta><t\<^isub>1>,v\<^isub>1,n\<^isub>1)"
-      proof -
-	have "n\<^isub>2\<sharp>v\<^isub>0" using ht' fresh_preserved fresh_psubst f by auto
-	then have "n\<^isub>2\<sharp>v\<^isub>0'" using eqc by auto
-	then have "n\<^isub>2\<sharp>((n\<^isub>1,v\<^isub>0')#\<theta>)" using f by (simp add: fresh_list_cons fresh_prod fresh_atm)
-	then have "n\<^isub>2\<sharp>((n\<^isub>1,v\<^isub>0')#\<theta>)<t\<^isub>1>" using f fresh_psubst by auto
-	moreover then have "n\<^isub>2 \<sharp> v\<^isub>1" using fresh_preserved ho by auto
-	ultimately show  "n\<^isub>2\<sharp>(\<theta><t'>,\<theta><t\<^isub>1>,v\<^isub>1,n\<^isub>1)" using f by (simp add: fresh_psubst fresh_atm)
-      qed
-    ultimately have "Case \<theta><t'> of inl n\<^isub>1 \<rightarrow> \<theta><t\<^isub>1> | inr n\<^isub>2 \<rightarrow> \<theta><t\<^isub>2> \<Down> v\<^isub>1 \<and> v\<^isub>1 \<in> V T" using ht' eqc by auto
-    moreover 
-      have "Case \<theta><t'> of inl n\<^isub>1 \<rightarrow> \<theta><t\<^isub>1> | inr n\<^isub>2 \<rightarrow> \<theta><t\<^isub>2> = \<theta><Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2>" 
-      using f by auto
-    ultimately have "\<exists>v. \<theta><Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2> \<Down> v \<and> v \<in> V T" by auto
-  }
-  moreover 
-  { fix v\<^isub>0'
-    assume eqc:"v\<^isub>0 = InR v\<^isub>0'" and "v\<^isub>0' \<in> V' S\<^isub>2"
-    then have inc:"v\<^isub>0' \<in> V (Data S\<^isub>2)" by auto
-    have "valid \<Gamma>" using th typing_implies_valid by auto
-    then moreover have "valid ((n\<^isub>2,Data S\<^isub>2)#\<Gamma>)" using f by auto
-    then moreover have "(n\<^isub>2,v\<^isub>0')#\<theta> Vcloses (n\<^isub>2,Data S\<^isub>2)#\<Gamma>" 
-      using inc h monotonicity by blast
-    moreover have ih:"\<And>\<Gamma> \<theta> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> t\<^isub>2 : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><t\<^isub>2> \<Down> v \<and> v \<in> V T" by fact
-    ultimately obtain v\<^isub>2 where ho:"((n\<^isub>2,v\<^isub>0')#\<theta>)<t\<^isub>2> \<Down> v\<^isub>2 \<and> v\<^isub>2 \<in> V T" using hr by blast
-    then have r:"\<theta><t\<^isub>2>[n\<^isub>2::=v\<^isub>0'] \<Down> v\<^isub>2 \<and> v\<^isub>2 \<in> V T" using psubst_subst_psubst f by simp
-    moreover have "n\<^isub>1\<sharp>(\<theta><t'>,\<theta><t\<^isub>2>,v\<^isub>2,n\<^isub>2)"
-    proof -
-      have "n\<^isub>1\<sharp>\<theta><t'>" using fresh_psubst f by simp
-      then have "n\<^isub>1\<sharp>v\<^isub>0" using ht' fresh_preserved by auto
-      then have "n\<^isub>1\<sharp>v\<^isub>0'" using eqc by auto
-      then have "n\<^isub>1\<sharp>((n\<^isub>2,v\<^isub>0')#\<theta>)" using f by (simp add: fresh_list_cons fresh_atm fresh_prod)
-      then have "n\<^isub>1\<sharp>((n\<^isub>2,v\<^isub>0')#\<theta>)<t\<^isub>2>" using f fresh_psubst by auto
-      moreover then have "n\<^isub>1\<sharp>v\<^isub>2" using fresh_preserved ho by auto
-      ultimately show  "n\<^isub>1 \<sharp> (\<theta><t'>,\<theta><t\<^isub>2>,v\<^isub>2,n\<^isub>2)" using f by (simp add: fresh_psubst fresh_atm)
-    qed
-    moreover have "n\<^isub>2 \<sharp> (\<theta><t'>,\<theta><t\<^isub>1>,v\<^isub>2,n\<^isub>1)"
-      proof -
-	have "n\<^isub>2\<sharp>\<theta><t'>" using fresh_psubst f by simp
-	then have "n\<^isub>2\<sharp>v\<^isub>0" using ht' fresh_preserved by auto
-	then have "n\<^isub>2\<sharp>v\<^isub>0'" using eqc by auto
-	then have "n\<^isub>2\<sharp>\<theta><t\<^isub>2>[n\<^isub>2::=v\<^isub>0']" using f fresh_subst_fresh by auto
-	then have "n\<^isub>2\<sharp>v\<^isub>2" using f fresh_preserved r by blast
-	then show "n\<^isub>2\<sharp>(\<theta><t'>,\<theta><t\<^isub>1>,v\<^isub>2,n\<^isub>1)" using f by (simp add: fresh_atm fresh_psubst)
-      qed
-    ultimately have "Case \<theta><t'> of inl n\<^isub>1 \<rightarrow> \<theta><t\<^isub>1> | inr n\<^isub>2 \<rightarrow> \<theta><t\<^isub>2> \<Down> v\<^isub>2 \<and> v\<^isub>2 \<in> V T" using ht' eqc by auto
-    then have "\<exists>v. \<theta><Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2> \<Down> v \<and> v \<in> V T" using f by auto
-  }
-  ultimately show "\<exists>v. \<theta><Case t' of inl n\<^isub>1 \<rightarrow> t\<^isub>1 | inr n\<^isub>2 \<rightarrow> t\<^isub>2> \<Down> v \<and> v \<in> V T" using hS V_sum by blast
-qed (force)+
+  case (Var x \<Gamma> \<theta> T)
+  have "\<Gamma> \<turnstile> (Var x) : T" by fact
+  then have "(x,T)\<in>set \<Gamma>" by (cases) (auto simp add: trm.inject)
+  with prems have "\<theta><Var x> \<Down> \<theta><Var x> \<and> \<theta><Var x>\<in> V T" 
+    by (auto intro!: Vs_reduce_to_themselves)
+  then show "\<exists>v. \<theta><Var x> \<Down> v \<and> v \<in> V T" by auto
+qed 
 
 theorem termination_of_evaluation:
   assumes a: "[] \<turnstile> e : T"
   shows "\<exists>v. e \<Down> v \<and> val v"
 proof -
-  from a have "\<exists>v. (([]::(name \<times> trm) list)<e>) \<Down> v \<and> v \<in> V T" 
-    by (rule termination_aux) (auto)
-  thus "\<exists>v. e \<Down> v \<and> val v" using V_are_values by auto
+  from a have "\<exists>v. []<e> \<Down> v \<and> v \<in> V T" by (rule termination_aux) (auto)
+  thus "\<exists>v. e \<Down> v \<and> val v" using Vs_are_values by auto
 qed 
 
 end