--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/Let_ExhaustIssue.thy	Fri Jul 01 17:46:15 2011 +0900
@@ -0,0 +1,130 @@
+theory Let
+imports "../Nominal2" 
+begin
+
+
+atom_decl name
+
+nominal_datatype trm =
+  Var "name"
+| App "trm" "trm"
+| Lam x::"name" t::"trm"  bind  x in t
+| Let as::"assn" t::"trm"   bind "bn as" in t
+and assn =
+  ANil
+| ACons "name" "trm" "assn"
+binder
+  bn
+where
+  "bn ANil = []"
+| "bn (ACons x t as) = (atom x) # (bn as)"
+
+lemma alpha_bn_inducts_raw:
+  "\<lbrakk>alpha_bn_raw a b; P3 ANil_raw ANil_raw;
+ \<And>trm_raw trm_rawa assn_raw assn_rawa name namea.
+    \<lbrakk>alpha_trm_raw trm_raw trm_rawa; alpha_bn_raw assn_raw assn_rawa;
+     P3 assn_raw assn_rawa\<rbrakk>
+    \<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw)
+        (ACons_raw namea trm_rawa assn_rawa)\<rbrakk> \<Longrightarrow> P3 a b"
+  by (erule alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.inducts(3)[of _ _ "\<lambda>x y. True" _ "\<lambda>x y. True", simplified]) auto
+
+lemmas alpha_bn_inducts = alpha_bn_inducts_raw[quot_lifted]
+
+lemma alpha_bn_refl: "alpha_bn x x"
+  by (induct x rule: trm_assn.inducts(2))
+     (rule TrueI, auto simp add: trm_assn.eq_iff)
+
+lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
+  by (simp add: permute_pure)
+
+lemma what_we_would_like:
+  assumes a: "alpha_bn as asa"
+  shows "\<forall>p. set (bn as) \<sharp>* fv_bn as \<and> set (bn asa) \<sharp>* fv_bn asa \<and>
+   p \<bullet> bn as = bn asa \<and> supp p \<subseteq> set (bn as) \<union> set (bn asa) \<longrightarrow> p \<bullet> as = asa"
+  apply (rule alpha_bn_inducts[OF a])
+  apply
+ (simp_all add: trm_assn.bn_defs trm_assn.perm_bn_simps trm_assn.supp)
+ apply clarify
+ apply simp
+ apply (simp add: atom_eqvt)
+ apply (case_tac "name = namea")
+ sorry
+
+lemma Abs_lst_fcb2:
+  fixes as bs :: "'a :: fs"
+    and x y :: "'b :: fs"
+    and c::"'c::fs"
+  assumes eq: "[ba as]lst. x = [ba bs]lst. y"
+  and fcb1: "set (ba as) \<sharp>* f as x c"
+  and fresh1: "set (ba as) \<sharp>* c"
+  and fresh2: "set (ba 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"
+  and ba_inj: "\<And>q r. q \<bullet> ba as = r \<bullet> ba bs \<Longrightarrow> q \<bullet> as = r \<bullet> bs"
+  shows "f as x c = f bs y c"
+sorry
+
+nominal_primrec
+    height_trm :: "trm \<Rightarrow> nat"
+and height_assn :: "assn \<Rightarrow> nat"
+where
+  "height_trm (Var x) = 1"
+| "height_trm (App l r) = max (height_trm l) (height_trm r)"
+| "height_trm (Lam v b) = 1 + (height_trm b)"
+| "set (bn as) \<sharp>* fv_bn as \<Longrightarrow> height_trm (Let as b) = max (height_assn as) (height_trm b)"
+| "height_assn ANil = 0"
+| "height_assn (ACons v t as) = max (height_trm t) (height_assn as)"
+  apply (simp only: eqvt_def height_trm_height_assn_graph_def)
+  apply (rule, perm_simp, rule, rule TrueI)
+  apply (case_tac x)
+  apply (rule_tac y="a" in trm_assn.strong_exhaust(1))
+  apply (auto)[4]
+  apply (drule_tac x="assn" in meta_spec)
+  apply (drule_tac x="trm" in meta_spec)
+  apply (simp add: alpha_bn_refl)
+--"HERE"
+  defer
+  apply (case_tac b rule: trm_assn.exhaust(2))
+  apply (auto)[2]
+  apply(simp_all)
+  apply (erule_tac c="()" in Abs_lst_fcb2)
+  apply (simp_all add: pure_fresh fresh_star_def)[3]
+  apply (simp add: eqvt_at_def)
+  apply (simp add: eqvt_at_def)
+  apply assumption
+  apply(erule conjE)
+  apply (simp add: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])
+  apply (simp add: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff])
+  apply (subgoal_tac "eqvt_at height_assn as")
+  apply (subgoal_tac "eqvt_at height_assn asa")
+  apply (subgoal_tac "eqvt_at height_trm b")
+  apply (subgoal_tac "eqvt_at height_trm ba")
+  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)")
+  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr asa)")
+  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl b)")
+  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl ba)")
+  defer
+  apply (simp add: eqvt_at_def height_trm_def)
+  apply (simp add: eqvt_at_def height_trm_def)
+  apply (simp add: eqvt_at_def height_assn_def)
+  apply (simp add: eqvt_at_def height_assn_def)
+  defer
+  apply (subgoal_tac "height_assn as = height_assn asa")
+  apply (subgoal_tac "height_trm b = height_trm ba")
+  apply simp
+  apply (erule_tac c="()" in Abs_lst_fcb2)
+  apply (simp_all add: pure_fresh fresh_star_def)[3]
+  apply (simp_all add: eqvt_at_def)[2]
+  apply assumption
+  apply (erule_tac Abs_lst_fcb)
+  apply (simp_all add: pure_fresh fresh_star_def)[2]
+  apply (drule what_we_would_like)
+  apply (drule_tac x="p" in spec)
+  apply simp
+  apply (simp add: eqvt_at_def)
+  oops
+
+end
+
+
+