--- a/Nominal/Term1.thy	Fri Mar 05 10:23:40 2010 +0100
+++ b/Nominal/Term1.thy	Fri Mar 05 13:04:47 2010 +0100
@@ -142,12 +142,16 @@
 apply default
 apply (rule rtrm1_bp_fs)+
 done
-
-lemma fv_eq_bv: "fv_bp bp = bv1 bp"
+lemma fv_eq_bv_pre: "fv_bp bp = bv1 bp"
 apply(induct bp rule: trm1_bp_inducts(2))
 apply(simp_all)
 done
 
+lemma fv_eq_bv: "fv_bp = bv1"
+apply(rule ext)
+apply(rule fv_eq_bv_pre)
+done
+
 lemma helper2: "{b. \<forall>pi. pi \<bullet> (a \<rightleftharpoons> b) \<bullet> bp \<noteq> bp} = {}"
 apply auto
 apply (rule_tac x="(x \<rightleftharpoons> a)" in exI)
@@ -165,6 +169,35 @@
 apply (rule alpha_bp_eq_eq)
 done
 
+lemma ex_out: 
+  "(\<exists>x. Z x \<and> Q) = (Q \<and> (\<exists>x. Z x))"
+  "(\<exists>x. Q \<and> Z x) = (Q \<and> (\<exists>x. Z x))"
+  "(\<exists>x. P x \<and> Q \<and> Z x) = (Q \<and> (\<exists>x. P x \<and> Z x))"
+  "(\<exists>x. Q \<and> P x \<and> Z x) = (Q \<and> (\<exists>x. P x \<and> Z x))"
+  "(\<exists>x. Q \<and> P x \<and> Z x \<and> W x) = (Q \<and> (\<exists>x. P x \<and> Z x \<and> W x))"
+apply (blast)+
+done
+
+lemma "(Abs bs (x, x') = Abs cs (y, y')) = (\<exists>p. (bs, x) \<approx>gen op = supp p (cs, y) \<and> (bs, x') \<approx>gen op = supp p (cs, y'))"
+thm Abs_eq_iff
+apply (simp add: Abs_eq_iff)
+apply (rule arg_cong[of _ _ "Ex"])
+apply (rule ext)
+apply (simp only: alpha_gen)
+apply (simp only: supp_Pair eqvts)
+apply rule
+apply (erule conjE)+
+oops
+
+lemma "(f (p \<bullet> bp), p \<bullet> bp) \<approx>gen op = f pi (f bp, bp) = False"
+apply (simp add: alpha_gen fresh_star_def)
+oops
+
+(* TODO: permute_ABS should be in eqvt? *)
+
+lemma Collect_neg_conj: "{x. \<not>(P x \<and> Q x)} = {x. \<not>(P x)} \<union> {x. \<not>(Q x)}"
+by (simp add: Collect_imp_eq Collect_neg_eq[symmetric])
+
 lemma supp_fv:
   "supp t = fv_trm1 t"
   "supp b = fv_bp b"
@@ -188,21 +221,23 @@
 apply(simp add: supp_at_base)
 apply(simp (no_asm) add: supp_def Collect_imp_eq Collect_neg_eq)
 apply(simp add: Collect_imp_eq[symmetric] Collect_neg_eq[symmetric] supp_def[symmetric])
-(*apply(subgoal_tac "supp (Lt1 bp rtrm11 rtrm12) = supp (Abs (bv1 bp) (rtrm12)) \<union> supp(rtrm11)*)
-apply(subgoal_tac "supp (Lt1 bp rtrm11 rtrm12) = supp (Abs (bv1 bp) (bp, rtrm12)) \<union> supp(rtrm11)")
+apply(subgoal_tac "supp (Lt1 bp rtrm11 rtrm12) = supp (Abs (bv1 bp) (rtrm12)) \<union> supp(rtrm11)")
+(*apply(subgoal_tac "supp (Lt1 bp rtrm11 rtrm12) = supp (Abs (bv1 bp) (bp, rtrm12)) \<union> supp(rtrm11)")*)
 apply(simp add: supp_Abs fv_trm1 supp_Pair Un_Diff Un_assoc fv_eq_bv)
 apply(blast) (* Un_commute in a good place *)
-apply(simp (no_asm) only: supp_def permute_trm1)
-apply(simp only: alpha1_INJ)
+apply(simp (no_asm) only: supp_def permute_set_eq atom_eqvt permute_trm1)
+apply(simp only: alpha1_INJ permute_ABS permute_prod.simps Abs_eq_iff)
+apply(simp only: ex_out)
 apply(simp only: Un_commute)
+apply(simp only: alpha_bp_eq fv_eq_bv)
+apply(simp only: alpha_gen fv_eq_bv supp_Pair)
+apply(simp only: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric] bv1_eqvt fv_eq_bv)
+apply(simp only: ex_out)
+apply(simp only: Collect_neg_conj finite_Un Diff_cancel)
+apply(simp)
 apply(simp add: Collect_imp_eq)
-apply(simp add: Collect_neg_eq[symmetric])
-apply(simp only: Abs_eq_iff)
-apply(simp add: alpha_bp_eq)
-apply(simp add: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric] bv1_eqvt fv_eq_bv)
-apply(simp add: alpha_gen fv_eq_bv supp_Pair)
-apply(simp add: supp_eqvt[symmetric] fv_trm1_eqvt[symmetric] bv1_eqvt fv_eq_bv)
-apply(simp add: Collect_imp_eq Collect_neg_eq Collect_disj_eq fresh_star_def helper2)
+apply(simp add: Collect_neg_eq[symmetric] fresh_star_def)
+apply(fold supp_def)
 sorry
 
 lemma trm1_supp: