--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Manual/Term8.thy	Tue Mar 23 08:19:33 2010 +0100
@@ -0,0 +1,61 @@
+theory Term8
+imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "Rsp" "../Attic/Prove"
+begin
+
+atom_decl name
+
+datatype rfoo8 =
+  Foo0 "name"
+| Foo1 "rbar8" "rfoo8" --"bind bv(bar) in foo"
+and rbar8 =
+  Bar0 "name"
+| Bar1 "name" "name" "rbar8" --"bind second name in b"
+
+primrec
+  rbv8
+where
+  "rbv8 (Bar0 x) = {}"
+| "rbv8 (Bar1 v x b) = {atom v}"
+
+setup {* snd o define_raw_perms (Datatype.the_info @{theory} "Term8.rfoo8") 2 *}
+print_theorems
+
+local_setup {* snd o define_fv_alpha (Datatype.the_info @{theory} "Term8.rfoo8") [
+  [[[]], [[], [(SOME @{term rbv8}, 0)]]], [[[]], [[], [(NONE, 1)], [(NONE, 1)]]]] *}
+notation
+  alpha_rfoo8 ("_ \<approx>f' _" [100, 100] 100) and
+  alpha_rbar8 ("_ \<approx>b' _" [100, 100] 100)
+thm alpha_rfoo8_alpha_rbar8.intros
+
+
+inductive
+  alpha8f :: "rfoo8 \<Rightarrow> rfoo8 \<Rightarrow> bool" ("_ \<approx>f _" [100, 100] 100)
+and
+  alpha8b :: "rbar8 \<Rightarrow> rbar8 \<Rightarrow> bool" ("_ \<approx>b _" [100, 100] 100)
+where
+  a1: "a = b \<Longrightarrow> (Foo0 a) \<approx>f (Foo0 b)"
+| a2: "a = b \<Longrightarrow> (Bar0 a) \<approx>b (Bar0 b)"
+| a3: "b1 \<approx>b b2 \<Longrightarrow> (\<exists>pi. (((rbv8 b1), t1) \<approx>gen alpha8f fv_rfoo8 pi ((rbv8 b2), t2))) \<Longrightarrow> Foo1 b1 t1 \<approx>f Foo1 b2 t2"
+| a4: "v1 = v2 \<Longrightarrow> (\<exists>pi. (({atom x1}, t1) \<approx>gen alpha8b fv_rbar8 pi ({atom x2}, t2))) \<Longrightarrow> Bar1 v1 x1 t1 \<approx>b Bar1 v2 x2 t2"
+
+lemma "(alpha8b ===> op =) rbv8 rbv8"
+  apply simp apply clarify
+  apply (erule alpha8f_alpha8b.inducts(2))
+  apply (simp_all)
+done
+
+lemma fv_rbar8_rsp_hlp: "x \<approx>b y \<Longrightarrow> fv_rbar8 x = fv_rbar8 y"
+  apply (erule alpha8f_alpha8b.inducts(2))
+  apply (simp_all add: alpha_gen)
+done
+lemma "(alpha8b ===> op =) fv_rbar8 fv_rbar8"
+  apply simp apply clarify apply (simp add: fv_rbar8_rsp_hlp)
+done
+
+lemma "(alpha8f ===> op =) fv_rfoo8 fv_rfoo8"
+  apply simp apply clarify
+  apply (erule alpha8f_alpha8b.inducts(1))
+  apply (simp_all add: alpha_gen fv_rbar8_rsp_hlp)
+done
+
+end