--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/Ex4.thy	Thu May 13 15:58:36 2010 +0100
@@ -0,0 +1,84 @@
+theory Ex4
+imports "../NewParser"
+begin
+
+declare [[STEPS = 4]]
+(* alpha does not work for this type *)
+
+atom_decl name
+
+nominal_datatype trm =
+  Var "name"
+| App "trm" "trm"
+| Lam x::"name" t::"trm"        bind_set x in t
+| Let p::"pat" "trm" t::"trm"   bind_set "f p" in t
+| Foo1 p::"pat" q::"pat" t::"trm" bind_set "f p" "f q" in t
+| Foo2 x::"name" p::"pat" t::"trm" bind_set x "f p" in t
+and pat =
+  PN
+| PS "name"
+| PD "pat" "pat"
+binder
+  f::"pat \<Rightarrow> atom set"
+where
+  "f PN = {}"
+| "f (PS x) = {atom x}"
+| "f (PD p1 p2) = (f p1) \<union> (f p2)"
+
+thm permute_trm_raw_permute_pat_raw.simps
+thm fv_trm_raw.simps fv_pat_raw.simps fv_f_raw.simps
+
+(* thm alpha_trm_raw_alpha_pat_raw_alpha_f_raw.intros[no_vars]*)
+
+inductive 
+ alpha_trm_raw and alpha_pat_raw and alpha_f_raw
+where
+(* alpha_trm_raw *)
+  "name = namea \<Longrightarrow> alpha_trm_raw (Var_raw name) (Var_raw namea)"
+| "\<lbrakk>alpha_trm_raw trm_raw1 trm_raw1a; alpha_trm_raw trm_raw2 trm_raw2a\<rbrakk>
+   \<Longrightarrow> alpha_trm_raw (App_raw trm_raw1 trm_raw2) (App_raw trm_raw1a trm_raw2a)"
+| "\<exists>p. ({atom name}, trm_raw) \<approx>gen alpha_trm_raw fv_trm_raw p ({atom namea}, trm_rawa) \<Longrightarrow>
+   alpha_trm_raw (Lam_raw name trm_raw) (Lam_raw namea trm_rawa)"
+| "\<lbrakk>\<exists>p. (f_raw pat_raw, trm_raw2) \<approx>gen alpha_trm_raw fv_trm_raw p (f_raw pat_rawa, trm_raw2a);
+   alpha_f_raw pat_raw pat_rawa; alpha_trm_raw trm_raw1 trm_raw1a\<rbrakk>
+  \<Longrightarrow> alpha_trm_raw (Let_raw pat_raw trm_raw1 trm_raw2) (Let_raw pat_rawa trm_raw1a trm_raw2a)"
+| "\<lbrakk>\<exists>p. (f_raw pat_raw1 \<union> f_raw pat_raw2, trm_raw) \<approx>gen alpha_trm_raw fv_trm_raw p (f_raw pat_raw1a \<union> f_raw pat_raw2a, trm_rawa);
+   alpha_f_raw pat_raw1 pat_raw1a; alpha_f_raw pat_raw2 pat_raw2a\<rbrakk>
+   \<Longrightarrow> alpha_trm_raw (Foo1_raw pat_raw1 pat_raw2 trm_raw) (Foo1_raw pat_raw1a pat_raw2a trm_rawa)"
+| "\<lbrakk>\<exists>p. ({atom name} \<union> f_raw pat_raw, trm_raw) \<approx>gen alpha_trm_raw fv_trm_raw p ({atom namea} \<union> f_raw pat_rawa, trm_rawa);
+   alpha_f_raw pat_raw pat_rawa\<rbrakk>
+   \<Longrightarrow> alpha_trm_raw (Foo2_raw name pat_raw trm_raw) (Foo2_raw namea pat_rawa trm_rawa)"
+
+| "alpha_pat_raw PN_raw PN_raw"
+| "name = namea \<Longrightarrow> alpha_pat_raw (PS_raw name) (PS_raw namea)"
+| "\<lbrakk>alpha_pat_raw pat_raw1 pat_raw1a; alpha_pat_raw pat_raw2 pat_raw2a\<rbrakk>
+   \<Longrightarrow> alpha_pat_raw (PD_raw pat_raw1 pat_raw2) (PD_raw pat_raw1a pat_raw2a)"
+
+| "alpha_f_raw PN_raw PN_raw"
+| "alpha_f_raw (PS_raw name) (PS_raw namea)"
+| "\<lbrakk>alpha_f_raw pat_raw1 pat_raw1a; alpha_f_raw pat_raw2 pat_raw2a\<rbrakk>
+  \<Longrightarrow> alpha_f_raw (PD_raw pat_raw1 pat_raw2) (PD_raw pat_raw1a pat_raw2a)"
+
+lemmas all = alpha_trm_raw_alpha_pat_raw_alpha_f_raw.intros
+
+lemma
+  shows "alpha_trm_raw (Foo2_raw x (PS_raw x) (Var_raw x))
+                       (Foo2_raw y (PS_raw y) (Var_raw y))"
+apply(rule all)
+apply(rule_tac x="(atom x \<rightleftharpoons> atom y)" in exI)
+apply(simp add: alphas)
+apply(simp add: supp_at_base fresh_star_def)
+apply(rule conjI)
+apply(rule all)
+apply(simp)
+apply(perm_simp)
+apply(simp)
+apply(rule all)
+done
+
+ 
+
+end
+
+
+