--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/ExLam.thy	Sat Apr 03 22:31:11 2010 +0200
@@ -0,0 +1,91 @@
+theory ExLam
+imports "../Parser"
+begin
+
+atom_decl name
+
+nominal_datatype lm =
+  Vr "name"
+| Ap "lm" "lm"
+| Lm x::"name" l::"lm"  bind x in l
+
+lemmas supp_fn' = lm.fv[simplified lm.supp]
+
+lemma
+  fixes c::"'a::fs"
+  assumes a1: "\<And>name c. P c (Vr name)"
+  and     a2: "\<And>lm1 lm2 c. \<lbrakk>\<And>d. P d lm1; \<And>d. P d lm2\<rbrakk> \<Longrightarrow> P c (Ap lm1 lm2)"
+  and     a3: "\<And>name lm c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lm\<rbrakk> \<Longrightarrow> P c (Lm name lm)"
+  shows "P c lm"
+proof -
+  have "\<And>p. P c (p \<bullet> lm)"
+    apply(induct lm arbitrary: c rule: lm.induct)
+    apply(simp only: lm.perm)
+    apply(rule a1)
+    apply(simp only: lm.perm)
+    apply(rule a2)
+    apply(blast)[1]
+    apply(assumption)
+    apply(subgoal_tac "\<exists>new::name. (atom new) \<sharp> (c, Lm (p \<bullet> name) (p \<bullet> lm))")
+    defer
+    apply(simp add: fresh_def)
+    apply(rule_tac X="supp (c, Lm (p \<bullet> name) (p \<bullet> lm))" in obtain_at_base)
+    apply(simp add: supp_Pair finite_supp)
+    apply(blast)
+    apply(erule exE)
+    apply(rule_tac t="p \<bullet> Lm name lm" and 
+                   s="(((p \<bullet> name) \<leftrightarrow> new) + p) \<bullet> Lm name lm" in subst)
+    apply(simp del: lm.perm)
+    apply(subst lm.perm)
+    apply(subst (2) lm.perm)
+    apply(rule flip_fresh_fresh)
+    apply(simp add: fresh_def)
+    apply(simp only: supp_fn')
+    apply(simp)
+    apply(simp add: fresh_Pair)
+    apply(simp)
+    apply(rule a3)
+    apply(simp add: fresh_Pair)
+    apply(drule_tac x="((p \<bullet> name) \<leftrightarrow> new) + p" in meta_spec)
+    apply(simp)
+    done
+  then have "P c (0 \<bullet> lm)" by blast
+  then show "P c lm" by simp
+qed
+
+
+lemma
+  fixes c::"'a::fs"
+  assumes a1: "\<And>name c. P c (Vr name)"
+  and     a2: "\<And>lm1 lm2 c. \<lbrakk>\<And>d. P d lm1; \<And>d. P d lm2\<rbrakk> \<Longrightarrow> P c (Ap lm1 lm2)"
+  and     a3: "\<And>name lm c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lm\<rbrakk> \<Longrightarrow> P c (Lm name lm)"
+  shows "P c lm"
+proof -
+  have "\<And>p. P c (p \<bullet> lm)"
+    apply(induct lm arbitrary: c rule: lm.induct)
+    apply(simp only: lm.perm)
+    apply(rule a1)
+    apply(simp only: lm.perm)
+    apply(rule a2)
+    apply(blast)[1]
+    apply(assumption)
+    thm at_set_avoiding
+    apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom name}) \<sharp>* c \<and> supp (p \<bullet> Lm name lm) \<sharp>* q")
+    apply(erule exE)
+    apply(rule_tac t="p \<bullet> Lm name lm" and 
+                   s="q \<bullet> p \<bullet> Lm name lm" in subst)
+    defer
+    apply(simp add: lm.perm)
+    apply(rule a3)
+    apply(simp add: eqvts fresh_star_def)
+    apply(drule_tac x="q + p" in meta_spec)
+    apply(simp)
+    sorry
+  then have "P c (0 \<bullet> lm)" by blast
+  then show "P c lm" by simp
+qed
+
+end
+
+
+