--- a/Nominal-General/Nominal2_Base.thy	Tue Apr 27 19:51:35 2010 +0200
+++ b/Nominal-General/Nominal2_Base.thy	Tue Apr 27 22:21:16 2010 +0200
@@ -1099,6 +1099,69 @@
     by auto
 qed
 
+
+section {* Concrete atoms types *}
+
+class at_base = pt +
+  fixes atom :: "'a \<Rightarrow> atom"
+  assumes atom_eq_iff [simp]: "atom a = atom b \<longleftrightarrow> a = b"
+  assumes atom_eqvt: "p \<bullet> (atom a) = atom (p \<bullet> a)"
+
+class at = at_base +
+  assumes sort_of_atom_eq [simp]: "sort_of (atom a) = sort_of (atom b)"
+
+lemma supp_at_base: 
+  fixes a::"'a::at_base"
+  shows "supp a = {atom a}"
+  by (simp add: supp_atom [symmetric] supp_def atom_eqvt)
+
+lemma fresh_at_base: 
+  shows "a \<sharp> b \<longleftrightarrow> a \<noteq> atom b"
+  unfolding fresh_def by (simp add: supp_at_base)
+
+instance at_base < fs
+proof qed (simp add: supp_at_base)
+
+lemma at_base_infinite [simp]:
+  shows "infinite (UNIV :: 'a::at_base set)" (is "infinite ?U")
+proof
+  obtain a :: 'a where "True" by auto
+  assume "finite ?U"
+  hence "finite (atom ` ?U)"
+    by (rule finite_imageI)
+  then obtain b where b: "b \<notin> atom ` ?U" "sort_of b = sort_of (atom a)"
+    by (rule obtain_atom)
+  from b(2) have "b = atom ((atom a \<rightleftharpoons> b) \<bullet> a)"
+    unfolding atom_eqvt [symmetric]
+    by (simp add: swap_atom)
+  hence "b \<in> atom ` ?U" by simp
+  with b(1) show "False" by simp
+qed
+
+lemma swap_at_base_simps [simp]:
+  fixes x y::"'a::at_base"
+  shows "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> x = y"
+  and   "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> y = x"
+  and   "atom x \<noteq> a \<Longrightarrow> atom x \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x"
+  unfolding atom_eq_iff [symmetric]
+  unfolding atom_eqvt [symmetric]
+  by simp_all
+
+lemma obtain_at_base:
+  assumes X: "finite X"
+  obtains a::"'a::at_base" where "atom a \<notin> X"
+proof -
+  have "inj (atom :: 'a \<Rightarrow> atom)"
+    by (simp add: inj_on_def)
+  with X have "finite (atom -` X :: 'a set)"
+    by (rule finite_vimageI)
+  with at_base_infinite have "atom -` X \<noteq> (UNIV :: 'a set)"
+    by auto
+  then obtain a :: 'a where "atom a \<notin> X"
+    by auto
+  thus ?thesis ..
+qed
+
 section {* library functions for the nominal infrastructure *}
 use "nominal_library.ML"