--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/ExLetRec.thy	Sat Apr 03 22:31:11 2010 +0200
@@ -0,0 +1,83 @@
+theory ExLetRec
+imports "../Parser"
+begin
+
+
+text {* example 3 or example 5 from Terms.thy *}
+
+atom_decl name
+
+ML {* val _ = recursive := true *}
+ML {* val _ = alpha_type := AlphaLst *}
+nominal_datatype trm =
+  Vr "name"
+| Ap "trm" "trm"
+| Lm x::"name" t::"trm"  bind x in t
+| Lt a::"lts" t::"trm"   bind "bn a" in t
+and lts =
+  Lnil
+| Lcons "name" "trm" "lts"
+binder
+  bn
+where
+  "bn Lnil = []"
+| "bn (Lcons x t l) = (atom x) # (bn l)"
+
+thm trm_lts.fv
+thm trm_lts.eq_iff
+thm trm_lts.bn
+thm trm_lts.perm
+thm trm_lts.induct
+thm trm_lts.distinct
+thm trm_lts.supp
+thm trm_lts.fv[simplified trm_lts.supp]
+
+(* why is this not in HOL simpset? *)
+lemma set_sub: "{a, b} - {b} = {a} - {b}"
+by auto
+
+lemma lets_bla:
+  "x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Lt (Lcons x (Vr y) Lnil) (Vr x)) \<noteq> (Lt (Lcons x (Vr z) Lnil) (Vr x))"
+  by (simp add: trm_lts.eq_iff alphas2 set_sub)
+
+lemma lets_ok:
+  "(Lt (Lcons x (Vr x) Lnil) (Vr x)) = (Lt (Lcons y (Vr y) Lnil) (Vr y))"
+  apply (simp add: trm_lts.eq_iff)
+  apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
+  apply (simp_all add: alphas2 fresh_star_def eqvts)
+  done
+
+lemma lets_ok3:
+  "x \<noteq> y \<Longrightarrow>
+   (Lt (Lcons x (Ap (Vr y) (Vr x)) (Lcons y (Vr y) Lnil)) (Ap (Vr x) (Vr y))) \<noteq>
+   (Lt (Lcons y (Ap (Vr x) (Vr y)) (Lcons x (Vr x) Lnil)) (Ap (Vr x) (Vr y)))"
+  apply (simp add: alphas trm_lts.eq_iff)
+  done
+
+
+lemma lets_not_ok1:
+  "x \<noteq> y \<Longrightarrow>
+   (Lt (Lcons x (Vr x) (Lcons y (Vr y) Lnil)) (Ap (Vr x) (Vr y))) \<noteq>
+   (Lt (Lcons y (Vr x) (Lcons x (Vr y) Lnil)) (Ap (Vr x) (Vr y)))"
+  apply (simp add: alphas trm_lts.eq_iff)
+  done
+
+lemma lets_nok:
+  "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>
+   (Lt (Lcons x (Ap (Vr z) (Vr z)) (Lcons y (Vr z) Lnil)) (Ap (Vr x) (Vr y))) \<noteq>
+   (Lt (Lcons y (Vr z) (Lcons x (Ap (Vr z) (Vr z)) Lnil)) (Ap (Vr x) (Vr y)))"
+  apply (simp add: alphas trm_lts.eq_iff fresh_star_def)
+  done
+
+lemma lets_ok4:
+  "(Lt (Lcons x (Ap (Vr y) (Vr x)) (Lcons y (Vr y) Lnil)) (Ap (Vr x) (Vr y))) =
+   (Lt (Lcons y (Ap (Vr x) (Vr y)) (Lcons x (Vr x) Lnil)) (Ap (Vr y) (Vr x)))"
+  apply (simp add: alphas trm_lts.eq_iff)
+  apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
+  apply (simp add: atom_eqvt fresh_star_def)
+  done
+
+end
+
+
+