
diff -r 7b5db6c23753 -r ca162f0a7957 Nominal/Ex/Lambda.thy
--- a/Nominal/Ex/Lambda.thy	Thu May 31 11:59:56 2012 +0100
+++ b/Nominal/Ex/Lambda.thy	Thu May 31 12:01:01 2012 +0100
@@ -299,13 +299,13 @@
 lemma forget:
   shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"
   by (nominal_induct t avoiding: x s rule: lam.strong_induct)
-     (auto simp add: lam.fresh fresh_at_base)
+     (auto simp add: fresh_at_base)
 
 text {* same lemma but with subst.induction *}
 lemma forget2:
   shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"
   by (induct t x s rule: subst.induct)
-     (auto simp add: lam.fresh fresh_at_base fresh_Pair)
+     (auto simp add:  fresh_at_base fresh_Pair)
 
 lemma fresh_fact:
   fixes z::"name"
@@ -314,7 +314,7 @@
   shows "atom z \<sharp> t[y ::= s]"
   using a b
   by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
-      (auto simp add: lam.fresh fresh_at_base)
+      (auto simp add:  fresh_at_base)
 
 lemma substitution_lemma:  
   assumes a: "x \<noteq> y" "atom x \<sharp> u"
@@ -328,7 +328,7 @@
   shows "t[x ::= s] = ((y \<leftrightarrow> x) \<bullet>t)[y ::= s]"
 using a 
 apply (nominal_induct t avoiding: x y s rule: lam.strong_induct)
-apply (auto simp add: lam.fresh fresh_at_base)
+apply (auto simp add:  fresh_at_base)
 done
 
 lemma height_ge_one:
@@ -368,7 +368,7 @@
 
 nominal_inductive beta
   avoids b4: "x"
-  by (simp_all add: fresh_star_def fresh_Pair lam.fresh fresh_fact)
+  by (simp_all add: fresh_star_def fresh_Pair  fresh_fact)
 
 text {* One-Reduction *}
 
@@ -385,7 +385,7 @@
 nominal_inductive One 
   avoids o3: "x"
       |  o4: "x"
-  by (simp_all add: fresh_star_def fresh_Pair lam.fresh fresh_fact)
+  by (simp_all add: fresh_star_def fresh_Pair  fresh_fact)
 
 lemma One_refl:
   shows "t \<longrightarrow>1 t"
@@ -610,7 +610,7 @@
 
 lemma var_fresh_subst:
   "atom x \<sharp> s \<Longrightarrow> atom x \<sharp> (t[x ::= s])"
-  by (induct t x s rule: subst.induct) (auto simp add: lam.supp lam.fresh fresh_at_base)
+  by (induct t x s rule: subst.induct) (auto simp add: lam.supp  fresh_at_base)
 
 (* function that evaluates a lambda term *)
 nominal_primrec
@@ -841,7 +841,7 @@
   apply(drule_tac x="namea" in meta_spec)
   apply(drule_tac x="lam" in meta_spec)
   apply(simp add: fresh_star_Pair)
-  apply(simp add: fresh_star_def fresh_at_base lam.fresh)
+  apply(simp add: fresh_star_def fresh_at_base )
   apply(auto)[1]
   apply(simp_all)[44]
   apply(simp del: Product_Type.prod.inject)  
@@ -880,7 +880,7 @@
   apply(drule_tac x="name" in meta_spec)
   apply(drule_tac x="lam" in meta_spec)
   apply(drule_tac x="(name \<leftrightarrow> namea) \<bullet> lama" in meta_spec)
-  apply(simp add: Abs1_eq_iff fresh_star_def fresh_Pair_elim fresh_at_base lam.fresh flip_def)
+  apply(simp add: Abs1_eq_iff fresh_star_def fresh_Pair_elim fresh_at_base flip_def)
   apply (metis Nominal2_Base.swap_commute fresh_permute_iff sort_of_atom_eq swap_atom_simps(2))
   apply simp_all
   apply (simp add: abs_same_binder)




