
diff -r 3c363a5070a5 -r bc86f5c3bc65 Nominal/Ex/Classical.thy
--- a/Nominal/Ex/Classical.thy	Tue Jun 28 00:30:30 2011 +0100
+++ b/Nominal/Ex/Classical.thy	Tue Jun 28 00:48:57 2011 +0100
@@ -46,6 +46,86 @@
 thm trm.supp
 thm trm.supp[simplified]
 
+lemma Abs_set_fcb2:
+  fixes as bs :: "atom set"
+    and x y :: "'b :: fs"
+    and c::"'c::fs"
+  assumes eq: "[as]set. x = [bs]set. 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]set. 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]set. x"]  
+      fin1 fin2 fin
+    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
+  have "[q \<bullet> as]set. (q \<bullet> x) = q \<bullet> ([as]set. x)" by simp
+  also have "\<dots> = [as]set. x"
+    by (simp only: fr2 perm_supp_eq)
+  finally have "[q \<bullet> as]set. (q \<bullet> x) = [bs]set. y" using eq by simp
+  then obtain r::perm where 
+    qq1: "q \<bullet> x = r \<bullet> y" and 
+    qq2: "q \<bullet> as = r \<bullet> bs" and 
+    qq3: "supp r \<subseteq> (q \<bullet> as) \<union> bs"
+    apply(drule_tac sym)
+    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)
+    apply(blast)
+    done
+  have "as \<sharp>* f as x c" by (rule fcb1)
+  then have "q \<bullet> (as \<sharp>* f as x c)"
+    by (simp add: permute_bool_def)
+  then have "(q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    apply(subst (asm) perm1)
+    using inc fresh1 fr1
+    apply(auto simp add: fresh_star_def fresh_Pair)
+    done
+  then have "(r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
+  then have "r \<bullet> (bs \<sharp>* f bs y c)"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    apply(subst (asm) perm2[symmetric])
+    using qq3 fresh2 fr1
+    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
+    done
+  then have fcb2: "bs \<sharp>* f bs y c" by (simp add: permute_bool_def)
+  have "f as x c = q \<bullet> (f as x c)"
+    apply(rule perm_supp_eq[symmetric])
+    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
+  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
+    apply(rule perm1)
+    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
+  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
+  also have "\<dots> = r \<bullet> (f bs y c)"
+    apply(rule perm2[symmetric])
+    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
+  also have "... = f bs y c"
+    apply(rule perm_supp_eq)
+    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
+  finally show ?thesis by simp
+qed
+
+
 lemma Abs_lst_fcb2:
   fixes as bs :: "atom list"
     and x y :: "'b :: fs"




