diff -r 1d9e50934bc5 -r aaef9dec5e1d Nominal/Ex/Let.thy
--- a/Nominal/Ex/Let.thy	Fri Jul 01 17:46:15 2011 +0900
+++ b/Nominal/Ex/Let.thy	Sat Jul 02 00:27:47 2011 +0100
@@ -39,8 +39,9 @@
 thm trm_assn.fresh
 thm trm_assn.exhaust
 thm trm_assn.strong_exhaust
+thm trm_assn.perm_bn_simps
 
-lemma alpha_bn_inducts_raw:
+lemma alpha_bn_inducts_raw[consumes 1]:
   "\<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;
@@ -49,7 +50,7 @@
         (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]
+lemmas alpha_bn_inducts[consumes 1] = alpha_bn_inducts_raw[quot_lifted]
 
 
 
@@ -66,6 +67,17 @@
   shows "bn x = bn y \<longrightarrow> x = y"
   by (rule alpha_bn_inducts[OF a]) (simp_all add: trm_assn.bn_defs)
 
+lemma bn_inj2:
+  assumes a: "alpha_bn x y"
+  shows "\<And>q r. (q \<bullet> bn x) = (r \<bullet> bn y) \<Longrightarrow> permute_bn q x = permute_bn r y"
+using a
+apply(induct rule: alpha_bn_inducts)
+apply(simp add: trm_assn.perm_bn_simps)
+apply(simp add: trm_assn.perm_bn_simps)
+apply(simp add: trm_assn.bn_defs)
+apply(simp add: atom_eqvt)
+done
+
 (*lemma alpha_bn_permute:
   assumes a: "alpha_bn x y"
       and b: "q \<bullet> bn x = r \<bullet> bn y"
@@ -85,6 +97,205 @@
     using bn_inj by simp
 *)
 
+
+lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
+  by (simp add: permute_pure)
+
+
+lemma Abs_lst_fcb2:
+  fixes as bs :: "'a :: fs"
+    and x y :: "'b :: fs"
+  assumes eq: "[ba as]lst. x = [ba bs]lst. y"
+  and ctxt: "finite (supp c)"
+  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"
+(* What we would like in this proof, and lets this proof finish *)
+  and ba_inj: "\<And>q r. q \<bullet> ba as = r \<bullet> ba bs \<Longrightarrow> pn q as = pn r bs"
+  shows "f as x c = f bs y c"
+proof -
+  have "supp (as, x, c) supports (f as x c)"
+    unfolding  supports_def fresh_def[symmetric]
+    apply (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
+    sorry
+  then have fin1: "finite (supp (f as x c))"
+    by (auto intro: supports_finite simp add: finite_supp supp_Pair ctxt)
+  have "supp (bs, y, c) supports (f bs y c)"
+    unfolding  supports_def fresh_def[symmetric]
+    apply (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
+    sorry
+  then have fin2: "finite (supp (f bs y c))"
+    by (auto intro: supports_finite simp add: finite_supp supp_Pair ctxt)
+  obtain q::"perm" where 
+    fr1: "(q \<bullet> (set (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and 
+    fr2: "supp q \<sharp>* ([ba as]lst. x)" and 
+    inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"
+    using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" 
+      and x="[ba as]lst. x"]  fin1 fin2
+    by (auto simp add: supp_Pair finite_supp ctxt Abs_fresh_star dest: fresh_star_supp_conv)
+  have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp
+  also have "\<dots> = [ba as]lst. x"
+    by (simp only: fr2 perm_supp_eq)
+  finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp
+  then obtain r::perm where 
+    qq1: "q \<bullet> x = r \<bullet> y" and 
+    qq2: "q \<bullet> (ba as) = r \<bullet> (ba bs)" and 
+    qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba bs)"
+    apply(drule_tac sym)
+    apply(simp only: Abs_eq_iff2 alphas)
+    apply(erule exE)
+    apply(erule conjE)+
+    apply(drule_tac x="p" in meta_spec)
+    apply(simp add: set_eqvt)
+    apply(blast)
+    done 
+  have "(set (ba as)) \<sharp>* f as x c" by (rule fcb1)
+  then have "q \<bullet> ((set (ba as)) \<sharp>* f as x c)"
+    by (simp add: permute_bool_def)
+  then have "set (q \<bullet> (ba as)) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    apply(subst (asm) perm1)
+    using inc fresh1 fr1
+    apply(auto simp add: fresh_star_def fresh_Pair)
+    done
+  then have "set (r \<bullet> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj
+    apply simp
+    sorry
+  then have "r \<bullet> ((set (ba bs)) \<sharp>* f bs y c)"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    apply(subst (asm) perm2[symmetric])
+    using qq3 fresh2 fr1
+    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
+    done
+  then have fcb2: "(set (ba bs)) \<sharp>* f bs y c" by (simp add: permute_bool_def)
+  have "f as x c = q \<bullet> (f as x c)"
+    apply(rule perm_supp_eq[symmetric])
+    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
+  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c"
+    apply(rule perm1)
+    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
+  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj
+    apply(simp)
+    sorry
+  also have "\<dots> = r \<bullet> (f bs y c)"
+    apply(rule perm2[symmetric])
+    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
+  also have "... = f bs y c"
+    apply(rule perm_supp_eq)
+    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
+  finally show ?thesis by simp
+qed
+
+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)"
+| "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 (case_tac a rule: trm_assn.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)
+  apply (case_tac b rule: trm_assn.exhaust(2))
+  apply (auto)[2]
+  apply(simp_all del: trm_assn.eq_iff)
+  apply(simp)
+  prefer 3
+  apply(simp)
+  apply(simp)
+  apply (erule_tac c="()" and pn="permute" in Abs_lst_fcb2)
+  apply(simp add: finite_supp)
+  apply (simp_all add: pure_fresh fresh_star_def)[3]
+  apply (simp add: eqvt_at_def)
+  apply (simp add: eqvt_at_def)
+  apply(auto)[1]
+  --"other case"
+  apply (simp only: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])
+  apply (simp only: 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)
+  apply (subgoal_tac "height_assn as = height_assn asa")
+  apply (subgoal_tac "height_trm b = height_trm ba")
+  apply simp
+  apply(simp)
+  apply(erule conjE)
+  apply (erule_tac c="()" and pn="permute_bn" in Abs_lst_fcb2)
+  apply(simp add: finite_supp)
+  apply (simp_all add: pure_fresh fresh_star_def)[3]
+  apply (simp_all add: eqvt_at_def)[2]
+  apply(simp add: bn_inj2)
+  apply(simp)
+  apply(erule conjE)
+  thm trm_assn.fv_defs
+  (*apply(simp add: Abs_eq_iff alphas)*)
+  apply (erule_tac c="()" and pn="permute_bn" and ba="bn" in Abs_lst_fcb2)
+  defer
+  apply (simp_all add: pure_fresh fresh_star_def)[3]
+  defer
+  defer
+  apply (simp_all add: eqvt_at_def)[2]
+  apply (rule bn_inj)
+  prefer 2
+  apply (simp add: eqvts)
+  oops
+
+nominal_primrec
+    subst  :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"
+and substa :: "name \<Rightarrow> trm \<Rightarrow> assn \<Rightarrow> assn"
+where
+  "subst s t (Var x) = (if (s = x) then t else (Var x))"
+| "subst s t (App l r) = App (subst s t l) (subst s t r)"
+| "atom v \<sharp> (s, t) \<Longrightarrow> subst s t (Lam v b) = Lam v (subst s t b)"
+| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (substa s t as) (subst s t b)"
+| "substa s t ANil = ANil"
+| "substa s t (ACons v t' as) = ACons v (subst v t t') as"
+(*unfolding eqvt_def subst_substa_graph_def
+  apply (rule, perm_simp)*)
+  defer
+  apply (rule TrueI)
+  apply (case_tac x)
+  apply (case_tac a)
+  apply (rule_tac y="c" and c="(aa,b)" in trm_assn.strong_exhaust(1))
+  apply (auto simp add: fresh_star_def)[3]
+  apply (drule_tac x="assn" in meta_spec)
+  apply (simp add: Abs1_eq_iff alpha_bn_refl)
+  apply (case_tac b)
+  apply (case_tac c rule: trm_assn.exhaust(2))
+  apply (auto)[2]
+  apply blast
+  apply blast
+  apply auto
+  apply (simp_all add: meta_eq_to_obj_eq[OF subst_def, symmetric, unfolded fun_eq_iff])
+  apply (simp_all add: meta_eq_to_obj_eq[OF substa_def, symmetric, unfolded fun_eq_iff])
+  prefer 2
+  apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)
+  apply (simp_all add: fresh_star_Pair)
+  prefer 6
+  apply (erule alpha_bn_inducts)
+ oops
+
+
 lemma lets_bla:
   "x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Let (ACons x (Var y) ANil) (Var x)) \<noteq> (Let (ACons x (Var z) ANil) (Var x))"
   by (simp add: trm_assn.eq_iff)
@@ -126,188 +337,4 @@
   apply (simp add: alphas trm_assn.supp supp_at_base x y fresh_star_def atom_eqvt)
   by (metis Rep_name_inverse atom_name_def flip_fresh_fresh fresh_atom fresh_perm x y)
 
-
-lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
-  by (simp add: permute_pure)
-
-
-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"
-(* What we would like in this proof, and lets this proof finish *)
-  and ba_inj: "\<And>q r. q \<bullet> ba as = r \<bullet> ba bs \<Longrightarrow> q \<bullet> as = r \<bullet> bs"
-(* What the user can supply with the help of alpha_bn *)
-(*  and bainj: "ba as = ba bs \<Longrightarrow> as = bs"*)
-  shows "f as x c = f bs y c"
-proof -
-  have "supp (as, x, c) supports (f as x c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin1: "finite (supp (f as x c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  have "supp (bs, y, c) supports (f bs y c)"
-    unfolding  supports_def fresh_def[symmetric]
-    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
-  then have fin2: "finite (supp (f bs y c))"
-    by (auto intro: supports_finite simp add: finite_supp)
-  obtain q::"perm" where 
-    fr1: "(q \<bullet> (set (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and 
-    fr2: "supp q \<sharp>* ([ba as]lst. x)" and 
-    inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"
-    using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" 
-      and x="[ba as]lst. x"]  fin1 fin2
-    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
-  have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp
-  also have "\<dots> = [ba as]lst. x"
-    by (simp only: fr2 perm_supp_eq)
-  finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp
-  then obtain r::perm where 
-    qq1: "q \<bullet> x = r \<bullet> y" and 
-    qq2: "q \<bullet> (ba as) = r \<bullet> (ba bs)" and 
-    qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba bs)"
-    apply(drule_tac sym)
-    apply(simp only: Abs_eq_iff2 alphas)
-    apply(erule exE)
-    apply(erule conjE)+
-    apply(drule_tac x="p" in meta_spec)
-    apply(simp add: set_eqvt)
-    apply(blast)
-    done
-  have "(set (ba as)) \<sharp>* f as x c" by (rule fcb1)
-  then have "q \<bullet> ((set (ba as)) \<sharp>* f as x c)"
-    by (simp add: permute_bool_def)
-  then have "set (q \<bullet> (ba as)) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm1)
-    using inc fresh1 fr1
-    apply(auto simp add: fresh_star_def fresh_Pair)
-    done
-  then have "set (r \<bullet> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj 
-    by simp
-  then have "r \<bullet> ((set (ba bs)) \<sharp>* f bs y c)"
-    apply(simp add: fresh_star_eqvt set_eqvt)
-    apply(subst (asm) perm2[symmetric])
-    using qq3 fresh2 fr1
-    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
-    done
-  then have fcb2: "(set (ba bs)) \<sharp>* f bs y c" by (simp add: permute_bool_def)
-  have "f as x c = q \<bullet> (f as x c)"
-    apply(rule perm_supp_eq[symmetric])
-    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c"
-    apply(rule perm1)
-    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
-  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 ba_inj by simp
-  also have "\<dots> = r \<bullet> (f bs y c)"
-    apply(rule perm2[symmetric])
-    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
-  also have "... = f bs y c"
-    apply(rule perm_supp_eq)
-    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
-  finally show ?thesis by simp
-qed
-
-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)"
-| "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 (case_tac a rule: trm_assn.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)
-  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)
-  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 c="()" and ba="bn" in Abs_lst_fcb2)
-  apply (simp_all add: pure_fresh fresh_star_def)[3]
-  apply (simp_all add: eqvt_at_def)[2]
-  apply (rule bn_inj)
-  prefer 2
-  apply (simp add: eqvts)
-  oops
-
-nominal_primrec
-    subst  :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"
-and substa :: "name \<Rightarrow> trm \<Rightarrow> assn \<Rightarrow> assn"
-where
-  "subst s t (Var x) = (if (s = x) then t else (Var x))"
-| "subst s t (App l r) = App (subst s t l) (subst s t r)"
-| "atom v \<sharp> (s, t) \<Longrightarrow> subst s t (Lam v b) = Lam v (subst s t b)"
-| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (substa s t as) (subst s t b)"
-| "substa s t ANil = ANil"
-| "substa s t (ACons v t' as) = ACons v (subst v t t') as"
-(*unfolding eqvt_def subst_substa_graph_def
-  apply (rule, perm_simp)*)
-  defer
-  apply (rule TrueI)
-  apply (case_tac x)
-  apply (case_tac a)
-  apply (rule_tac y="c" and c="(aa,b)" in trm_assn.strong_exhaust(1))
-  apply (auto simp add: fresh_star_def)[3]
-  apply (drule_tac x="assn" in meta_spec)
-  apply (simp add: Abs1_eq_iff alpha_bn_refl)
-  apply (case_tac b)
-  apply (case_tac c rule: trm_assn.exhaust(2))
-  apply (auto)[2]
-  apply blast
-  apply blast
-  apply auto
-  apply (simp_all add: meta_eq_to_obj_eq[OF subst_def, symmetric, unfolded fun_eq_iff])
-  apply (simp_all add: meta_eq_to_obj_eq[OF substa_def, symmetric, unfolded fun_eq_iff])
-  prefer 2
-  apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)
-  apply (simp_all add: fresh_star_Pair)
-  prefer 6
-  apply (erule alpha_bn_inducts)
- oops
-
-
-end
-
-
-
+end
\ No newline at end of file