--- a/Nominal/Ex/SubstNoFcb.thy	Tue Feb 19 05:38:46 2013 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,84 +0,0 @@
-theory Lambda imports "../Nominal2" begin
-
-atom_decl name
-
-nominal_datatype lam =
-  Var "name"
-| App "lam" "lam"
-| Lam x::"name" l::"lam"  binds x in l ("Lam [_]. _" [100, 100] 100)
-
-nominal_primrec lam_rec ::
-  "(name \<Rightarrow> 'a :: pt) \<Rightarrow> (lam \<Rightarrow> lam \<Rightarrow> 'a) \<Rightarrow> (name \<Rightarrow> lam \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b :: fs \<Rightarrow> lam \<Rightarrow> 'a"
-where
-  "lam_rec fv fa fl fd P (Var n) = fv n"
-| "lam_rec fv fa fl fd P (App l r) = fa l r"
-| "(atom x \<sharp> P \<and> (\<forall>y s. atom y \<sharp> P \<longrightarrow> Lam [x]. t = Lam [y]. s \<longrightarrow> fl x t = fl y s)) \<Longrightarrow>
-     lam_rec fv fa fl fd P (Lam [x]. t) = fl x t"
-| "(atom x \<sharp> P \<and> \<not>(\<forall>y s. atom y \<sharp> P \<longrightarrow> Lam [x]. t = Lam [y]. s \<longrightarrow> fl x t = fl y s)) \<Longrightarrow>
-     lam_rec fv fa fl fd P (Lam [x]. t) = fd"
-  apply (simp add: eqvt_def lam_rec_graph_def)
-  apply (rule, perm_simp, rule, rule)
-  apply (case_tac x)
-  apply (rule_tac y="f" and c="e" in lam.strong_exhaust)
-  apply metis
-  apply metis
-  unfolding fresh_star_def
-  apply simp
-  apply metis
-  apply simp_all[2]
-  apply (metis (no_types) Pair_inject lam.distinct)+
-  apply simp
-  apply (metis (no_types) Pair_inject lam.distinct)+
-  done
-
-termination (eqvt) by lexicographic_order
-
-lemma lam_rec_cong[fundef_cong]:
-  " (\<And>v. l = Var v \<Longrightarrow> fv v = fv' v) \<Longrightarrow>
-    (\<And>t1 t2. l = App t1 t2 \<Longrightarrow> fa t1 t2 = fa' t1 t2) \<Longrightarrow>
-    (\<And>n t. l = Lam [n]. t \<Longrightarrow> fl n t = fl' n t) \<Longrightarrow>
-  lam_rec fv fa fl fd P l = lam_rec fv' fa' fl' fd P l"
-  apply (rule_tac y="l" and c="P" in lam.strong_exhaust)
-  apply auto
-  apply (case_tac "(\<forall>y s. atom y \<sharp> P \<longrightarrow> Lam [name]. lam = Lam [y]. s \<longrightarrow> fl name lam = fl y s)")
-  apply (subst lam_rec.simps) apply (simp add: fresh_star_def)
-  apply (subst lam_rec.simps) apply (simp add: fresh_star_def)
-  using Abs1_eq_iff lam.eq_iff apply metis
-  apply (subst lam_rec.simps(4)) apply (simp add: fresh_star_def)
-  apply (subst lam_rec.simps(4)) apply (simp add: fresh_star_def)
-  using Abs1_eq_iff lam.eq_iff apply metis
-  done
-
-nominal_primrec substr where
-[simp del]: "substr l y s = lam_rec
-  (%x. if x = y then s else (Var x))
-  (%t1 t2. App (substr t1 y s) (substr t2 y s))
-  (%x t. Lam [x]. (substr t y s)) (Lam [y]. Var y) (y, s) l"
-unfolding eqvt_def substr_graph_def
-apply (rule, perm_simp, rule, rule)
-by pat_completeness auto
-
-termination (eqvt) by lexicographic_order
-
-lemma fresh_fun_eqvt_app3:
-  assumes e: "eqvt f"
-  shows "\<lbrakk>a \<sharp> x; a \<sharp> y; a \<sharp> z\<rbrakk> \<Longrightarrow> a \<sharp> f x y z"
-  using fresh_fun_eqvt_app[OF e] fresh_fun_app by (metis (lifting, full_types))
-
-lemma substr_simps:
-  "substr (Var x) y s = (if x = y then s else (Var x))"
-  "substr (App t1 t2) y s = App (substr t1 y s) (substr t2 y s)"
-  "atom x \<sharp> (y, s) \<Longrightarrow> substr (Lam [x]. t) y s = Lam [x]. (substr t y s)"
-  apply (subst substr.simps) apply (simp only: lam_rec.simps)
-  apply (subst substr.simps) apply (simp only: lam_rec.simps)
-  apply (subst substr.simps) apply (subst lam_rec.simps)
-  apply (auto simp add: Abs1_eq_iff substr.eqvt swap_fresh_fresh)
-  apply (rule fresh_fun_eqvt_app3[of substr])
-  apply (simp add: eqvt_def eqvts_raw)
-  apply (simp_all add: fresh_Pair)
-  done
-
-end
-
-
-