--- a/Nominal/Ex/Let.thy	Sat Jul 02 00:27:47 2011 +0100
+++ b/Nominal/Ex/Let.thy	Sat Jul 02 12:40:59 2011 +0900
@@ -2,7 +2,6 @@
 imports "../Nominal2" 
 begin
 
-
 atom_decl name
 
 nominal_datatype trm =
@@ -78,71 +77,43 @@
 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"
-    shows "alpha_bn (q \<bullet> x) (r \<bullet> y)"
-proof -
-  have "alpha_bn x (permute_bn r y)"
-    by (rule alpha_bn_trans[OF a]) (rule trm_assn.perm_bn_alpha)
-  then have "alpha_bn (permute_bn r y) x"
-    by (rule alpha_bn_sym)
-  then have "alpha_bn (permute_bn r y) (permute_bn q x)"
-    by (rule alpha_bn_trans) (rule trm_assn.perm_bn_alpha)
-  then have "alpha_bn (permute_bn q x) (permute_bn r y)"
-    by (rule alpha_bn_sym)
-  moreover have "bn (permute_bn q x) = bn (permute_bn r y)"
-    using b trm_assn.permute_bn by simp
-  ultimately have "permute_bn q x = permute_bn r y"
-    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"
+  fixes as bs :: "atom list"
     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 c::"'c::fs"
+  assumes eq: "[as]lst. x = [bs]lst. y"
+  and fcb1: "(set as) \<sharp>* c \<Longrightarrow> (set as) \<sharp>* f as x c"
+  and fresh1: "set as \<sharp>* c"
+  and fresh2: "set 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
+    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 supp_Pair ctxt)
+    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]
-    apply (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
-    sorry
+    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 supp_Pair ctxt)
+    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 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"
+    fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and 
+    fr2: "supp q \<sharp>* Abs_lst as x" and 
+    inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"
+    using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]  
+      fin1 fin2
+    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
+  have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp
+  also have "\<dots> = Abs_lst as 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
+  finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs 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)"
+    qq2: "q \<bullet> as = r \<bullet> bs" and 
+    qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"
     apply(drule_tac sym)
     apply(simp only: Abs_eq_iff2 alphas)
     apply(erule exE)
@@ -150,35 +121,34 @@
     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)"
+    done
+  have "(set as) \<sharp>* f as x c" 
+    apply(rule fcb1)
+    apply(rule fresh1)
+    done
+  then have "q \<bullet> ((set 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"
+  then have "set (q \<bullet> 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)"
+  then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp
+  then have "r \<bullet> ((set 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)
+  then have fcb2: "(set 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"
+    using inc fcb1[OF fresh1] 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> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 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)
@@ -188,119 +158,146 @@
   finally show ?thesis by simp
 qed
 
+lemma Abs_lst1_fcb2:
+  fixes a b :: "atom"
+    and x y :: "'b :: fs"
+    and c::"'c :: fs"
+  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"
+  and fcb1: "a \<sharp> c \<Longrightarrow> a \<sharp> f a x c"
+  and fresh: "{a, b} \<sharp>* c"
+  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"
+  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"
+  shows "f a x c = f b y c"
+using e
+apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])
+apply(simp_all)
+using fcb1 fresh perm1 perm2
+apply(simp_all add: fresh_star_def)
+done
+
+
+lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"
+  by (simp add: permute_pure)
+
+function
+  apply_assn :: "(trm \<Rightarrow> nat) \<Rightarrow> assn \<Rightarrow> nat"
+where
+  "apply_assn f ANil = (0 :: nat)"
+| "apply_assn f (ACons x t as) = max (f t) (apply_assn f as)"
+apply(case_tac x)
+apply(case_tac b rule: trm_assn.exhaust(2))
+apply(simp_all)
+apply(blast)
+done
+
+termination by lexicographic_order
+
+lemma [eqvt]:
+  "p \<bullet> (apply_assn f a) = apply_assn (p \<bullet> f) (p \<bullet> a)"
+  apply(induct f a rule: apply_assn.induct)
+  apply simp_all
+  apply(perm_simp)
+  apply rule
+  apply(perm_simp)
+  apply simp
+  done
+
+lemma alpha_bn_apply_assn:
+  assumes "alpha_bn as bs"
+  shows "apply_assn f as = apply_assn f bs"
+  using assms
+  apply (induct rule: alpha_bn_inducts)
+  apply simp_all
+  done
+
 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)
+| "height_trm (Let as b) = max (apply_assn height_trm as) (height_trm b)"
+  apply (simp only: eqvt_def height_trm_graph_def)
   apply (rule, perm_simp, rule, rule TrueI)
-  apply (case_tac x)
-  apply (case_tac a rule: trm_assn.exhaust(1))
+  apply (case_tac x 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)
+  apply (erule_tac c="()" in Abs_lst1_fcb2)
+  apply (simp_all add: pure_fresh fresh_star_def eqvt_at_def)[4]
+  apply (erule conjE)
+  apply (subst alpha_bn_apply_assn)
+  apply assumption
+  apply (rule arg_cong) back
+  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(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
+  done
+
+definition "height_assn = apply_assn height_trm"
+
+function
+  apply_assn2 :: "(trm \<Rightarrow> trm) \<Rightarrow> assn \<Rightarrow> assn"
+where
+  "apply_assn2 f ANil = ANil"
+| "apply_assn2 f (ACons x t as) = ACons x (f t) (apply_assn2 f as)"
+  apply(case_tac x)
+  apply(case_tac b rule: trm_assn.exhaust(2))
+  apply(simp_all)
+  apply(blast)
+  done
+
+termination by lexicographic_order
+
+lemma [eqvt]:
+  "p \<bullet> (apply_assn2 f a) = apply_assn2 (p \<bullet> f) (p \<bullet> a)"
+  apply(induct f a rule: apply_assn2.induct)
+  apply simp_all
+  apply(perm_simp)
+  apply rule
+  done
+
+lemma bn_apply_assn2: "bn (apply_assn2 f as) = bn as"
+  apply (induct as rule: trm_assn.inducts(2))
+  apply (rule TrueI)
+  apply (simp_all add: trm_assn.bn_defs)
+  done
 
 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
+| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (apply_assn2 (subst s t) as) (subst s t b)"
+  apply (simp only: eqvt_def subst_graph_def)
+  apply (rule, perm_simp, rule)
   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 (rule_tac y="c" and c="(a,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_lst1_fcb2)
+  apply (simp add: Abs_fresh_iff)
+  apply (simp add: fresh_star_def)
+  apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def)[2]
+  apply (simp add: bn_apply_assn2)
   apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)
-  apply (simp_all add: fresh_star_Pair)
-  prefer 6
+  apply (simp add: fresh_star_def Abs_fresh_iff)
+  apply assumption+
+  apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def trm_assn.fv_bn_eqvt)[2]
   apply (erule alpha_bn_inducts)
- oops
-
+  apply simp_all
+  done
 
 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)
 
-
 lemma lets_ok:
   "(Let (ACons x (Var y) ANil) (Var x)) = (Let (ACons y (Var y) ANil) (Var y))"
   apply (simp add: trm_assn.eq_iff Abs_eq_iff )
@@ -337,4 +334,4 @@
   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)
 
-end
\ No newline at end of file
+end