diff --git a/src/HOL/Nominal/Examples/SOS.thy b/src/HOL/Nominal/Examples/SOS.thy
--- a/src/HOL/Nominal/Examples/SOS.thy
+++ b/src/HOL/Nominal/Examples/SOS.thy
@@ -1,552 +1,549 @@
 (*                                                        *)
 (* Formalisation of some typical SOS-proofs.              *)
 (*                                                        *) 
 (* This work was inspired by 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 "HOL-Nominal.Nominal"
 begin
 
 atom_decl name
 
 text \<open>types and terms\<close>
 nominal_datatype ty = 
     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"
 
 lemma fresh_ty:
   fixes x::"name" 
   and   T::"ty"
   shows "x\<sharp>T"
 by (induct T rule: ty.induct)
    (auto simp add: fresh_nat)
 
 text \<open>Parallel and single substitution.\<close>
 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[eqvt]:
   fixes pi::"name prm"
   shows "pi\<bullet>(lookup \<theta> X) = lookup (pi\<bullet>\<theta>) (pi\<bullet>X)"
 by (induct \<theta>) (auto simp add: eqvts)
 
 lemma lookup_fresh:
   fixes z::"name"
   assumes a: "z\<sharp>\<theta>" and b: "z\<sharp>x"
   shows "z \<sharp>lookup \<theta> x"
 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)
 
 (* parallel substitution *)
 
 nominal_primrec
   psubst :: "(name\<times>trm) list \<Rightarrow> trm \<Rightarrow> trm"  ("_<_>" [95,95] 105)
 where
   "\<theta><(Var x)> = (lookup \<theta> x)"
 | "\<theta><(App e\<^sub>1 e\<^sub>2)> = App (\<theta><e\<^sub>1>) (\<theta><e\<^sub>2>)"
 | "x\<sharp>\<theta> \<Longrightarrow> \<theta><(Lam [x].e)> = Lam [x].(\<theta><e>)"
-apply(finite_guess)+
-apply(rule TrueI)+
-apply(simp add: abs_fresh)+
-apply(fresh_guess)+
-done
+  by (finite_guess | simp add: abs_fresh | fresh_guess)+
 
 lemma psubst_eqvt[eqvt]:
   fixes pi::"name prm" 
   and   t::"trm"
   shows "pi\<bullet>(\<theta><t>) = (pi\<bullet>\<theta>)<(pi\<bullet>t)>"
 by (nominal_induct t avoiding: \<theta> rule: trm.strong_induct)
    (perm_simp add: fresh_bij lookup_eqvt)+
 
 lemma fresh_psubst: 
   fixes z::"name"
   and   t::"trm"
   assumes "z\<sharp>t" and "z\<sharp>\<theta>"
   shows "z\<sharp>(\<theta><t>)"
 using assms
 by (nominal_induct t avoiding: z \<theta> t rule: trm.strong_induct)
    (auto simp add: abs_fresh lookup_fresh)
 
 lemma psubst_empty[simp]:
   shows "[]<t> = t"
   by (nominal_induct t rule: trm.strong_induct) 
      (auto simp add: fresh_list_nil)
 
 text \<open>Single substitution\<close>
 abbreviation 
   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\<^sub>1 t\<^sub>2)[y::=t'] = App (t\<^sub>1[y::=t']) (t\<^sub>2[y::=t'])"
   and   "x\<sharp>(y,t') \<Longrightarrow> (Lam [x].t)[y::=t'] = Lam [x].(t[y::=t'])"
 by (simp_all add: fresh_list_cons fresh_list_nil)
 
 lemma fresh_subst:
   fixes z::"name"
   shows "\<lbrakk>z\<sharp>s; (z=y \<or> z\<sharp>t)\<rbrakk> \<Longrightarrow> z\<sharp>t[y::=s]"
 by (nominal_induct t avoiding: z y s rule: trm.strong_induct)
    (auto simp add: abs_fresh fresh_prod fresh_atm)
 
 lemma forget: 
   assumes a: "x\<sharp>e" 
   shows "e[x::=e'] = e"
 using a
 by (nominal_induct e avoiding: x e' rule: trm.strong_induct)
    (auto simp add: fresh_atm abs_fresh)
 
 lemma psubst_subst_psubst:
   assumes h: "x\<sharp>\<theta>"
   shows "\<theta><e>[x::=e'] = ((x,e')#\<theta>)<e>"
   using h
 by (nominal_induct e avoiding: \<theta> x e' rule: trm.strong_induct)
    (auto simp add: fresh_list_cons fresh_atm forget lookup_fresh lookup_fresh')
 
 text \<open>Typing Judgements\<close>
 
 inductive
   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_elim[elim]: "valid ((x,T)#\<Gamma>)"
 
 lemma valid_insert:
   assumes a: "valid (\<Delta>@[(x,T)]@\<Gamma>)"
   shows "valid (\<Delta> @ \<Gamma>)" 
 using a
 by (induct \<Delta>)
    (auto simp add:  fresh_list_append fresh_list_cons elim!: valid_elim)
 
 lemma fresh_set: 
   shows "y\<sharp>xs = (\<forall>x\<in>set xs. y\<sharp>x)"
 by (induct xs) (simp_all add: fresh_list_nil fresh_list_cons)
 
 lemma context_unique:
   assumes a1: "valid \<Gamma>"
   and     a2: "(x,T) \<in> set \<Gamma>"
   and     a3: "(x,U) \<in> set \<Gamma>"
   shows "T = U" 
 using a1 a2 a3
 by (induct) (auto simp add: fresh_set fresh_prod fresh_atm)
 
 text \<open>Typing Relation\<close>
 
 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\<^sub>1 : T\<^sub>1\<rightarrow>T\<^sub>2; \<Gamma> \<turnstile> e\<^sub>2 : T\<^sub>1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App e\<^sub>1 e\<^sub>2 : T\<^sub>2"
 | t_Lam[intro]:   "\<lbrakk>x\<sharp>\<Gamma>; (x,T\<^sub>1)#\<Gamma> \<turnstile> e : T\<^sub>2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].e : T\<^sub>1\<rightarrow>T\<^sub>2"
 
 equivariance typing
 
 nominal_inductive typing
   by (simp_all add: abs_fresh fresh_ty)
 
 lemma typing_implies_valid:
   assumes a: "\<Gamma> \<turnstile> t : T"
   shows "valid \<Gamma>"
 using a by (induct) (auto)
 
 
 lemma t_App_elim:
   assumes a: "\<Gamma> \<turnstile> App t1 t2 : T"
   obtains T' where "\<Gamma> \<turnstile> t1 : T' \<rightarrow> T" and "\<Gamma> \<turnstile> t2 : T'"
 using a
 by (cases) (auto simp add: trm.inject)
 
 lemma t_Lam_elim: 
   assumes a: "\<Gamma> \<turnstile> Lam [x].t : T" "x\<sharp>\<Gamma>"
   obtains T\<^sub>1 and T\<^sub>2 where "(x,T\<^sub>1)#\<Gamma> \<turnstile> t : T\<^sub>2" and "T=T\<^sub>1\<rightarrow>T\<^sub>2"
 using a
 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>\<^sub>1 \<subseteq> \<Gamma>\<^sub>2 \<equiv> \<forall>x T. (x,T)\<in>set \<Gamma>\<^sub>1 \<longrightarrow> (x,T)\<in>set \<Gamma>\<^sub>2"
 
 lemma weakening: 
   fixes \<Gamma>\<^sub>1 \<Gamma>\<^sub>2::"(name\<times>ty) list"
   assumes "\<Gamma>\<^sub>1 \<turnstile> e: T" and "valid \<Gamma>\<^sub>2" and "\<Gamma>\<^sub>1 \<subseteq> \<Gamma>\<^sub>2"
   shows "\<Gamma>\<^sub>2 \<turnstile> e: T"
   using assms
 proof(nominal_induct \<Gamma>\<^sub>1 e T avoiding: \<Gamma>\<^sub>2 rule: typing.strong_induct)
   case (t_Lam x \<Gamma>\<^sub>1 T\<^sub>1 t T\<^sub>2 \<Gamma>\<^sub>2)
   have vc: "x\<sharp>\<Gamma>\<^sub>2" by fact
   have ih: "\<lbrakk>valid ((x,T\<^sub>1)#\<Gamma>\<^sub>2); (x,T\<^sub>1)#\<Gamma>\<^sub>1 \<subseteq> (x,T\<^sub>1)#\<Gamma>\<^sub>2\<rbrakk> \<Longrightarrow> (x,T\<^sub>1)#\<Gamma>\<^sub>2 \<turnstile> t : T\<^sub>2" by fact
   have "valid \<Gamma>\<^sub>2" by fact
   then have "valid ((x,T\<^sub>1)#\<Gamma>\<^sub>2)" using vc by auto
   moreover
   have "\<Gamma>\<^sub>1 \<subseteq> \<Gamma>\<^sub>2" by fact
   then have "(x,T\<^sub>1)#\<Gamma>\<^sub>1 \<subseteq> (x,T\<^sub>1)#\<Gamma>\<^sub>2" by simp
   ultimately have "(x,T\<^sub>1)#\<Gamma>\<^sub>2 \<turnstile> t : T\<^sub>2" using ih by simp 
   with vc show "\<Gamma>\<^sub>2 \<turnstile> Lam [x].t : T\<^sub>1\<rightarrow>T\<^sub>2" by auto
 qed (auto)
 
 lemma type_substitutivity_aux:
   assumes a: "(\<Delta>@[(x,T')]@\<Gamma>) \<turnstile> e : T"
   and     b: "\<Gamma> \<turnstile> e' : T'"
   shows "(\<Delta>@\<Gamma>) \<turnstile> e[x::=e'] : T" 
 using a b 
 proof (nominal_induct \<Gamma>\<equiv>"\<Delta>@[(x,T')]@\<Gamma>" e T avoiding: e' \<Delta> rule: typing.strong_induct)
   case (t_Var y T e' \<Delta>)
   then have a1: "valid (\<Delta>@[(x,T')]@\<Gamma>)" 
        and  a2: "(y,T) \<in> set (\<Delta>@[(x,T')]@\<Gamma>)" 
        and  a3: "\<Gamma> \<turnstile> e' : T'" .
   from a1 have a4: "valid (\<Delta>@\<Gamma>)" by (rule valid_insert)
   { assume eq: "x=y"
     from a1 a2 have "T=T'" using eq by (auto intro: context_unique)
     with a3 have "\<Delta>@\<Gamma> \<turnstile> Var y[x::=e'] : T" using eq a4 by (auto intro: weakening)
   }
   moreover
   { assume ineq: "x\<noteq>y"
     from a2 have "(y,T) \<in> set (\<Delta>@\<Gamma>)" using ineq by simp
     then have "\<Delta>@\<Gamma> \<turnstile> Var y[x::=e'] : T" using ineq a4 by auto
   }
   ultimately show "\<Delta>@\<Gamma> \<turnstile> Var y[x::=e'] : T" by blast
 qed (force simp add: fresh_list_append fresh_list_cons)+
 
 corollary type_substitutivity:
   assumes a: "(x,T')#\<Gamma> \<turnstile> e : T"
   and     b: "\<Gamma> \<turnstile> e' : T'"
   shows "\<Gamma> \<turnstile> e[x::=e'] : T"
 using a b type_substitutivity_aux[where \<Delta>="[]"]
 by (auto)
 
 text \<open>Values\<close>
 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\<^sub>1 e\<^sub>2)" 
   "\<not> val (Var x)"
   by (auto elim: val.cases)
 
 text \<open>Big-Step Evaluation\<close>
 
 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\<^sub>1,e\<^sub>2,e'); e\<^sub>1\<Down>Lam [x].e; e\<^sub>2\<Down>e\<^sub>2'; e[x::=e\<^sub>2']\<Down>e'\<rbrakk> \<Longrightarrow> App e\<^sub>1 e\<^sub>2 \<Down> e'"
 
 equivariance big
 
 nominal_inductive big
   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 a by (induct) (auto simp add: abs_fresh fresh_subst)
 
 lemma b_App_elim:
   assumes a: "App e\<^sub>1 e\<^sub>2 \<Down> e'" "x\<sharp>(e\<^sub>1,e\<^sub>2,e')"
   obtains f\<^sub>1 and f\<^sub>2 where "e\<^sub>1 \<Down> Lam [x]. f\<^sub>1" "e\<^sub>2 \<Down> f\<^sub>2" "f\<^sub>1[x::=f\<^sub>2] \<Down> e'"
   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"
   shows "\<Gamma> \<turnstile> e' : T"
   using a b
 proof (nominal_induct avoiding: \<Gamma> arbitrary: T rule: big.strong_induct) 
   case (b_App x e\<^sub>1 e\<^sub>2 e' e e\<^sub>2' \<Gamma> T)
   have vc: "x\<sharp>\<Gamma>" by fact
   have "\<Gamma> \<turnstile> App e\<^sub>1 e\<^sub>2 : T" by fact
   then obtain T' where a1: "\<Gamma> \<turnstile> e\<^sub>1 : T'\<rightarrow>T" and a2: "\<Gamma> \<turnstile> e\<^sub>2 : T'" 
     by (cases) (auto simp add: trm.inject)
   have ih1: "\<Gamma> \<turnstile> e\<^sub>1 : T' \<rightarrow> T \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].e : T' \<rightarrow> T" by fact
   have ih2: "\<Gamma> \<turnstile> e\<^sub>2 : T' \<Longrightarrow> \<Gamma> \<turnstile> e\<^sub>2' : T'" by fact 
   have ih3: "\<Gamma> \<turnstile> e[x::=e\<^sub>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 elim: t_Lam_elim simp add: ty.inject)
   moreover
   have "\<Gamma> \<turnstile> e\<^sub>2': T'" using ih2 a2 by simp
   ultimately have "\<Gamma> \<turnstile> e[x::=e\<^sub>2'] : T" by (simp add: type_substitutivity)
   thus "\<Gamma> \<turnstile> e' : T" using ih3 by simp
 qed (blast)
 
 lemma subject_reduction2:
   assumes a: "e \<Down> e'" and b: "\<Gamma> \<turnstile> e : T"
   shows "\<Gamma> \<turnstile> e' : T"
   using a b
 by (nominal_induct avoiding: \<Gamma> T rule: big.strong_induct)
    (force elim: t_App_elim t_Lam_elim simp add: ty.inject type_substitutivity)+
 
 lemma unicity_of_evaluation:
   assumes a: "e \<Down> e\<^sub>1" 
   and     b: "e \<Down> e\<^sub>2"
   shows "e\<^sub>1 = e\<^sub>2"
   using a b
 proof (nominal_induct e e\<^sub>1 avoiding: e\<^sub>2 rule: big.strong_induct)
   case (b_Lam x e t\<^sub>2)
   have "Lam [x].e \<Down> t\<^sub>2" by fact
   thus "Lam [x].e = t\<^sub>2" by cases (simp_all add: trm.inject)
 next
   case (b_App x e\<^sub>1 e\<^sub>2 e' e\<^sub>1' e\<^sub>2' t\<^sub>2)
   have ih1: "\<And>t. e\<^sub>1 \<Down> t \<Longrightarrow> Lam [x].e\<^sub>1' = t" by fact
   have ih2:"\<And>t. e\<^sub>2 \<Down> t \<Longrightarrow> e\<^sub>2' = t" by fact
   have ih3: "\<And>t. e\<^sub>1'[x::=e\<^sub>2'] \<Down> t \<Longrightarrow> e' = t" by fact
   have app: "App e\<^sub>1 e\<^sub>2 \<Down> t\<^sub>2" by fact
   have vc: "x\<sharp>e\<^sub>1" "x\<sharp>e\<^sub>2" "x\<sharp>t\<^sub>2" by fact+
   then have "x\<sharp>App e\<^sub>1 e\<^sub>2" by auto
   from app vc obtain f\<^sub>1 f\<^sub>2 where x1: "e\<^sub>1 \<Down> Lam [x]. f\<^sub>1" and x2: "e\<^sub>2 \<Down> f\<^sub>2" and x3: "f\<^sub>1[x::=f\<^sub>2] \<Down> t\<^sub>2"
     by (auto elim!: b_App_elim)
   then have "Lam [x]. f\<^sub>1 = Lam [x]. e\<^sub>1'" using ih1 by simp
   then 
   have "f\<^sub>1 = e\<^sub>1'" by (auto simp add: trm.inject alpha) 
   moreover 
   have "f\<^sub>2 = e\<^sub>2'" using x2 ih2 by simp
   ultimately have "e\<^sub>1'[x::=e\<^sub>2'] \<Down> t\<^sub>2" using x3 by simp
   thus "e' = t\<^sub>2" using ih3 by simp
 qed
 
 lemma reduces_evaluates_to_values:
   assumes h: "t \<Down> t'"
   shows "val t'"
 using h by (induct) (auto)
 
 (* Valuation *)
 
 nominal_primrec
   V :: "ty \<Rightarrow> trm set" 
 where
   "V (TVar x) = {e. val e}"
 | "V (T\<^sub>1 \<rightarrow> T\<^sub>2) = {Lam [x].e | x e. \<forall> v \<in> (V T\<^sub>1). \<exists> v'. e[x::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2}"
   by (rule TrueI)+ 
 
 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.strong_induct)
-apply(auto simp add: trm.inject)
-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(force)
-apply(auto)
-apply(rule_tac x="pi\<bullet>v'" in exI)
-apply(auto)
-apply(drule_tac pi="pi" in big.eqvt)
-apply(perm_simp add: eqvts)
-done
+  assumes "x \<in> V T"
+  shows "(pi\<bullet>x) \<in> V T"
+using assms
+proof (nominal_induct T arbitrary: pi x rule: ty.strong_induct)
+  case (TVar nat)
+  then show ?case
+    by (simp add: val.eqvt)
+next
+  case (Arrow T\<^sub>1 T\<^sub>2 pi x)
+  obtain a e where ae: "x = Lam [a]. e" "\<forall>v\<in>V T\<^sub>1. \<exists>v'. e[a::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2"
+    using Arrow.prems by auto
+  have "\<exists>v'. (pi \<bullet> e)[(pi \<bullet> a)::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2" if v: "v \<in> V T\<^sub>1" for v
+  proof -
+    have "rev pi \<bullet> v \<in> V T\<^sub>1"
+      by (simp add: Arrow.hyps(1) v)
+    then obtain v' where "e[a::=(rev pi \<bullet> v)] \<Down> v'" "v' \<in> V T\<^sub>2"
+      using ae(2) by blast
+    then have "(pi \<bullet> e)[(pi \<bullet> a)::=v] \<Down> pi \<bullet> v'"
+      by (metis (no_types, lifting) big.eqvt cons_eqvt nil_eqvt perm_pi_simp(1) perm_prod.simps psubst_eqvt)
+    then show ?thesis
+      using Arrow.hyps \<open>v' \<in> V T\<^sub>2\<close> by blast
+  qed
+  with ae show ?case by force
+qed
 
 lemma V_arrow_elim_weak:
   assumes h:"u \<in> V (T\<^sub>1 \<rightarrow> T\<^sub>2)"
   obtains a t where "u = Lam [a].t" and "\<forall> v \<in> (V T\<^sub>1). \<exists> v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2"
 using h by (auto)
 
 lemma V_arrow_elim_strong:
   fixes c::"'a::fs_name"
   assumes h: "u \<in> V (T\<^sub>1 \<rightarrow> T\<^sub>2)"
   obtains a t where "a\<sharp>c" "u = Lam [a].t" "\<forall>v \<in> (V T\<^sub>1). \<exists> v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2"
-using h
-apply -
-apply(erule V_arrow_elim_weak)
-apply(subgoal_tac "\<exists>a'::name. a'\<sharp>(a,t,c)") (*A*)
-apply(erule exE)
-apply(drule_tac x="a'" in meta_spec)
-apply(drule_tac x="[(a,a')]\<bullet>t" in meta_spec)
-apply(drule meta_mp)
-apply(simp)
-apply(drule meta_mp)
-apply(simp add: trm.inject alpha fresh_left fresh_prod calc_atm fresh_atm)
-apply(perm_simp)
-apply(force)
-apply(drule meta_mp)
-apply(rule ballI)
-apply(drule_tac x="[(a,a')]\<bullet>v" in bspec)
-apply(simp add: V_eqvt)
-apply(auto)
-apply(rule_tac x="[(a,a')]\<bullet>v'" in exI)
-apply(auto)
-apply(drule_tac pi="[(a,a')]" in big.eqvt)
-apply(perm_simp add: eqvts calc_atm)
-apply(simp add: V_eqvt)
-(*A*)
-apply(rule exists_fresh')
-apply(simp add: fin_supp)
-done
+proof -
+  obtain a t where "u = Lam [a].t" 
+    and at: "\<And>v. v \<in> (V T\<^sub>1) \<Longrightarrow> \<exists> v'. t[a::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2"
+    using V_arrow_elim_weak [OF assms] by metis
+  obtain a'::name where a': "a'\<sharp>(a,t,c)"
+    by (meson exists_fresh fs_name_class.axioms)
+  then have "u = Lam [a'].([(a, a')] \<bullet> t)"
+    unfolding \<open>u = Lam [a].t\<close>
+    by (smt (verit) alpha fresh_atm fresh_prod perm_swap trm.inject(2))
+  moreover
+  have "\<exists> v'. ([(a, a')] \<bullet> t)[a'::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2" if "v \<in> (V T\<^sub>1)" for v
+  proof -
+    obtain v' where v': "t[a::=([(a, a')] \<bullet> v)] \<Down> v' \<and> v' \<in> V T\<^sub>2"
+      using V_eqvt \<open>v \<in> V T\<^sub>1\<close> at by blast
+    then have "([(a, a')] \<bullet> t[a::=([(a, a')] \<bullet> v)]) \<Down> [(a, a')] \<bullet> v'"
+      by (simp add: big.eqvt)
+    then show ?thesis
+      by (smt (verit) V_eqvt cons_eqvt nil_eqvt perm_prod.simps perm_swap(1) psubst_eqvt swap_simps(1) v')
+  qed
+  ultimately show thesis
+    by (metis fresh_prod that a')
+qed
 
 lemma Vs_are_values:
   assumes a: "e \<in> V T"
   shows "val e"
 using a by (nominal_induct T arbitrary: e rule: ty.strong_induct) (auto)
 
 lemma values_reduce_to_themselves:
   assumes a: "val v"
   shows "v \<Down> v"
 using a by (induct) (auto)
 
 lemma Vs_reduce_to_themselves:
   assumes a: "v \<in> V T"
   shows "v \<Down> v"
 using a by (simp add: values_reduce_to_themselves Vs_are_values)
 
 text \<open>'\<theta> maps x to e' asserts that \<theta> substitutes x with e\<close>
 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"
 
 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)"
 
 lemma case_distinction_on_context:
   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
 qed
 
 lemma monotonicity:
   fixes m::"name"
   fixes \<theta>::"(name \<times> trm) list" 
   assumes h1: "\<theta> Vcloses \<Gamma>"
   and     h2: "e \<in> V T" 
   and     h3: "valid ((x,T)#\<Gamma>)"
   shows "(x,e)#\<theta> Vcloses (x,T)#\<Gamma>"
 proof(intro strip)
   fix x' T'
   assume "(x',T') \<in> set ((x,T)#\<Gamma>)"
   then have "((x',T')=(x,T)) \<or> ((x',T')\<in>set \<Gamma> \<and> x'\<noteq>x)" using h3 
     by (rule_tac case_distinction_on_context)
   moreover
   { (* first case *)
     assume "(x',T') = (x,T)"
     then have "\<exists>e'. ((x,e)#\<theta>) maps x to e' \<and> e' \<in> V T'" using h2 by auto
   }
   moreover
   { (* second case *)
     assume "(x',T') \<in> set \<Gamma>" and neq:"x' \<noteq> x"
       then have "\<exists>e'. \<theta> maps x' to e' \<and> e' \<in> V T'" using h1 by auto
       then have "\<exists>e'. ((x,e)#\<theta>) maps x' to e' \<and> e' \<in> V T'" using neq by auto
   }
   ultimately show "\<exists>e'.  ((x,e)#\<theta>) maps x' to e'  \<and> e' \<in> V T'" by blast
 qed
 
 lemma termination_aux:
   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.strong_induct)
   case (App e\<^sub>1 e\<^sub>2 \<Gamma> \<theta> T)
   have ih\<^sub>1: "\<And>\<theta> \<Gamma> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> e\<^sub>1 : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><e\<^sub>1> \<Down> v \<and> v \<in> V T" by fact
   have ih\<^sub>2: "\<And>\<theta> \<Gamma> T. \<lbrakk>\<theta> Vcloses \<Gamma>; \<Gamma> \<turnstile> e\<^sub>2 : T\<rbrakk> \<Longrightarrow> \<exists>v. \<theta><e\<^sub>2> \<Down> v \<and> v \<in> V T" by fact
   have as\<^sub>1: "\<theta> Vcloses \<Gamma>" by fact 
   have as\<^sub>2: "\<Gamma> \<turnstile> App e\<^sub>1 e\<^sub>2 : T" by fact
   then obtain T' where "\<Gamma> \<turnstile> e\<^sub>1 : T' \<rightarrow> T" and "\<Gamma> \<turnstile> e\<^sub>2 : T'" by (auto elim: t_App_elim)
   then obtain v\<^sub>1 v\<^sub>2 where "(i)": "\<theta><e\<^sub>1> \<Down> v\<^sub>1" "v\<^sub>1 \<in> V (T' \<rightarrow> T)"
                       and "(ii)": "\<theta><e\<^sub>2> \<Down> v\<^sub>2" "v\<^sub>2 \<in> V T'" using ih\<^sub>1 ih\<^sub>2 as\<^sub>1 by blast
   from "(i)" obtain x e' 
             where "v\<^sub>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\<^sub>1> \<Down> Lam [x].e'" 
             and fr: "x\<sharp>(\<theta>,e\<^sub>1,e\<^sub>2)" by (blast elim: V_arrow_elim_strong)
   from fr have fr\<^sub>1: "x\<sharp>\<theta><e\<^sub>1>" and fr\<^sub>2: "x\<sharp>\<theta><e\<^sub>2>" by (simp_all add: fresh_psubst)
   from "(ii)" "(iii)" obtain v\<^sub>3 where "(v)": "e'[x::=v\<^sub>2] \<Down> v\<^sub>3" "v\<^sub>3 \<in> V T" by auto
   from fr\<^sub>2 "(ii)" have "x\<sharp>v\<^sub>2" by (simp add: big_preserves_fresh)
   then have "x\<sharp>e'[x::=v\<^sub>2]" by (simp add: fresh_subst)
   then have fr\<^sub>3: "x\<sharp>v\<^sub>3" using "(v)" by (simp add: big_preserves_fresh)
   from fr\<^sub>1 fr\<^sub>2 fr\<^sub>3 have "x\<sharp>(\<theta><e\<^sub>1>,\<theta><e\<^sub>2>,v\<^sub>3)" by simp
   with "(iv)" "(ii)" "(v)" have "App (\<theta><e\<^sub>1>) (\<theta><e\<^sub>2>) \<Down> v\<^sub>3" by auto
   then show "\<exists>v. \<theta><App e\<^sub>1 e\<^sub>2> \<Down> v \<and> v \<in> V T" using "(v)" 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\<^sub>1: "\<theta> Vcloses \<Gamma>" by fact
   have as\<^sub>2: "\<Gamma> \<turnstile> Lam [x].e : T" by fact
   have fs: "x\<sharp>\<Gamma>" "x\<sharp>\<theta>" by fact+
   from as\<^sub>2 fs obtain T\<^sub>1 T\<^sub>2 
     where "(i)": "(x,T\<^sub>1)#\<Gamma> \<turnstile> e:T\<^sub>2" and "(ii)": "T = T\<^sub>1 \<rightarrow> T\<^sub>2" using fs
     by (auto elim: t_Lam_elim)
   from "(i)" have "(iii)": "valid ((x,T\<^sub>1)#\<Gamma>)" by (simp add: typing_implies_valid)
   have "\<forall>v \<in> (V T\<^sub>1). \<exists>v'. (\<theta><e>)[x::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2"
   proof
     fix v
     assume "v \<in> (V T\<^sub>1)"
     with "(iii)" as\<^sub>1 have "(x,v)#\<theta> Vcloses (x,T\<^sub>1)#\<Gamma>" using monotonicity by auto
     with ih "(i)" obtain v' where "((x,v)#\<theta>)<e> \<Down> v' \<and> v' \<in> V T\<^sub>2" by blast
     then have "\<theta><e>[x::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2" using fs by (simp add: psubst_subst_psubst)
     then show "\<exists>v'. \<theta><e>[x::=v] \<Down> v' \<and> v' \<in> V T\<^sub>2" by auto
   qed
   then have "Lam[x].\<theta><e> \<in> V (T\<^sub>1 \<rightarrow> T\<^sub>2)" by auto
   then have "\<theta><Lam [x].e> \<Down> Lam [x].\<theta><e> \<and> Lam [x].\<theta><e> \<in> V (T\<^sub>1\<rightarrow>T\<^sub>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 (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 Var 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. []<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