
diff -r bc86f5c3bc65 -r db0786a521fd Nominal/Ex/Classical.thy
--- a/Nominal/Ex/Classical.thy	Tue Jun 28 00:48:57 2011 +0100
+++ b/Nominal/Ex/Classical.thy	Tue Jun 28 12:36:34 2011 +0900
@@ -125,6 +125,64 @@
   finally show ?thesis by simp
 qed
 
+lemma Abs_res_fcb2:
+  fixes as bs :: "atom set"
+    and x y :: "'b :: fs"
+    and c::"'c::fs"
+  assumes eq: "[as]res. x = [bs]res. y"
+  and fin: "finite as" "finite bs"
+  and fcb1: "as \<sharp>* f as x c"
+  and fresh1: "as \<sharp>* c"
+  and fresh2: "bs \<sharp>* c"
+  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
+  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
+  shows "f as x c = f bs y c"
+proof -
+  have "supp (as, x, c) supports (f as x c)"
+    unfolding  supports_def fresh_def[symmetric]
+    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
+  then have fin1: "finite (supp (f as x c))"
+    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
+  have "supp (bs, y, c) supports (f bs y c)"
+    unfolding  supports_def fresh_def[symmetric]
+    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
+  then have fin2: "finite (supp (f bs y c))"
+    using fin by (auto intro: supports_finite simp add: finite_supp supp_of_finite_sets supp_Pair)
+  obtain q::"perm" where 
+    fr1: "(q \<bullet> as) \<sharp>* (x, c, f as x c, f bs y c)" and 
+    fr2: "supp q \<sharp>* ([as]res. x)" and 
+    inc: "supp q \<subseteq> as \<union> (q \<bullet> as)"
+    using at_set_avoiding3[where xs="as" and c="(x, c, f as x c, f bs y c)" and x="[as]res. x"]  
+      fin1 fin2 fin
+    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
+  have "[q \<bullet> as]res. (q \<bullet> x) = q \<bullet> ([as]res. x)" by simp
+  also have "\<dots> = [as]res. x"
+    by (simp only: fr2 perm_supp_eq)
+  finally have "[q \<bullet> as]res. (q \<bullet> x) = [bs]res. y" using eq by simp
+  then obtain r::perm where 
+    qq1: "q \<bullet> x = r \<bullet> y" and 
+    qq2: "(q \<bullet> as \<inter> supp (q \<bullet> x)) = r \<bullet> (bs \<inter> supp y)" and 
+    qq3: "supp r \<subseteq> bs \<inter> supp y \<union> q \<bullet> as \<inter> supp (q \<bullet> x)"
+    apply(drule_tac sym)
+    apply(subst(asm) Abs_eq_res_set)
+    apply(simp only: Abs_eq_iff2 alphas)
+    apply(erule exE)
+    apply(erule conjE)+
+    apply(drule_tac x="p" in meta_spec)
+    apply(simp add: set_eqvt)
+    done
+  have "(as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c" sorry (* FCB? *)
+  then have "q \<bullet> ((as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c)"
+    by (simp add: permute_bool_def)
+  then have "(q \<bullet> (as \<inter> supp x)) \<sharp>* f (q \<bullet> (as \<inter> supp x)) (q \<bullet> x) c"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    sorry (* perm? *)
+  then have "r \<bullet> (bs \<inter> supp y) \<sharp>* f (r \<bullet> (bs \<inter> supp y)) (r \<bullet> y) c" using qq2 apply (simp add: inter_eqvt)
+  (* rest similar reversing it other way around... *)
+  show ?thesis sorry
+qed
+
+
 
 lemma Abs_lst_fcb2:
   fixes as bs :: "atom list"
@@ -229,6 +287,10 @@
   shows "p \<bullet> l = l \<longleftrightarrow> supp p \<inter> set l = {}"
   by (induct l) (auto simp add: supp_Nil supp_perm)
 
+lemma permute_length_eq:
+  shows "p \<bullet> xs = ys \<Longrightarrow> length xs = length ys"
+  by (auto simp add: length_eqvt[symmetric] permute_pure)
+
 lemma Abs_lst_binder_length:
   shows "[xs]lst. T = [ys]lst. S \<Longrightarrow> length xs = length ys"
   by (auto simp add: Abs_eq_iff alphas length_eqvt[symmetric] permute_pure)
@@ -239,6 +301,13 @@
      (metis fresh_star_zero inf_absorb1 permute_atom_list_id supp_perm_eq
        supp_zero_perm_zero)
 
+lemma in_permute_list:
+  shows "py \<bullet> p \<bullet> xs = px \<bullet> xs \<Longrightarrow>  x \<in> set xs \<Longrightarrow> py \<bullet> p \<bullet> x = px \<bullet> x"
+  by (induct xs) auto
+
+
+
+
 nominal_primrec 
   crename :: "trm \<Rightarrow> coname \<Rightarrow> coname \<Rightarrow> trm"  ("_[_\<turnstile>c>_]" [100,100,100] 100) 
 where




