theory Lambda
imports 
  "../Nominal2"
  "~~/src/HOL/Library/Monad_Syntax"
begin


atom_decl name

nominal_datatype lam =
  Var "name"
| App "lam" "lam"
| Lam x::"name" l::"lam"  binds x in l ("Lam [_]. _" [100, 100] 100)


section {* Simple examples from Norrish 2004 *}

nominal_primrec 
  is_app :: "lam \<Rightarrow> bool"
where
  "is_app (Var x) = False"
| "is_app (App t1 t2) = True"
| "is_app (Lam [x]. t) = False"
apply(simp add: eqvt_def is_app_graph_def)
apply (rule, perm_simp, rule)
apply(rule TrueI)
apply(rule_tac y="x" in lam.exhaust)
apply(auto)[3]
apply(all_trivials)
done

termination (eqvt) by lexicographic_order

thm is_app_def
thm is_app.eqvt

thm eqvts

nominal_primrec 
  rator :: "lam \<Rightarrow> lam option"
where
  "rator (Var x) = None"
| "rator (App t1 t2) = Some t1"
| "rator (Lam [x]. t) = None"
apply(simp add: eqvt_def rator_graph_def)
apply (rule, perm_simp, rule)
apply(rule TrueI)
apply(rule_tac y="x" in lam.exhaust)
apply(auto)[3]
apply(simp_all)
done

termination (eqvt) by lexicographic_order

nominal_primrec 
  rand :: "lam \<Rightarrow> lam option"
where
  "rand (Var x) = None"
| "rand (App t1 t2) = Some t2"
| "rand (Lam [x]. t) = None"
apply(simp add: eqvt_def rand_graph_def)
apply (rule, perm_simp, rule)
apply(rule TrueI)
apply(rule_tac y="x" in lam.exhaust)
apply(auto)[3]
apply(simp_all)
done

termination (eqvt) by lexicographic_order

nominal_primrec 
  is_eta_nf :: "lam \<Rightarrow> bool"
where
  "is_eta_nf (Var x) = True"
| "is_eta_nf (App t1 t2) = (is_eta_nf t1 \<and> is_eta_nf t2)"
| "is_eta_nf (Lam [x]. t) = (is_eta_nf t \<and> 
                             ((is_app t \<and> rand t = Some (Var x)) \<longrightarrow> atom x \<in> supp (rator t)))"
apply(simp add: eqvt_def is_eta_nf_graph_def)
apply (rule, perm_simp, rule)
apply(rule TrueI)
apply(rule_tac y="x" in lam.exhaust)
apply(auto)[3]
apply(simp_all)
apply(erule_tac c="()" in Abs_lst1_fcb2')
apply(simp_all add: pure_fresh fresh_star_def)[3]
apply(simp add: eqvt_at_def conj_eqvt)
apply(perm_simp)
apply(rule refl)
apply(simp add: eqvt_at_def conj_eqvt)
apply(perm_simp)
apply(rule refl)
done

termination (eqvt) by lexicographic_order

nominal_datatype path = Left | Right | In

section {* Paths to a free variables *} 

instance path :: pure
apply(default)
apply(induct_tac "x::path" rule: path.induct)
apply(simp_all)
done

nominal_primrec 
  var_pos :: "name \<Rightarrow> lam \<Rightarrow> (path list) set"
where
  "var_pos y (Var x) = (if y = x then {[]} else {})"
| "var_pos y (App t1 t2) = (Cons Left ` (var_pos y t1)) \<union> (Cons Right ` (var_pos y t2))"
| "atom x \<sharp> y \<Longrightarrow> var_pos y (Lam [x]. t) = (Cons In ` (var_pos y t))"
apply(simp add: eqvt_def var_pos_graph_def)
apply (rule, perm_simp, rule)
apply(rule TrueI)
apply(case_tac x)
apply(rule_tac y="b" and c="a" in lam.strong_exhaust)
apply(auto simp add: fresh_star_def)[3]
apply(simp_all)
apply(erule conjE)+
apply(erule_tac Abs_lst1_fcb2)
apply(simp add: pure_fresh)
apply(simp add: fresh_star_def)
apply(simp add: eqvt_at_def image_eqvt perm_supp_eq)
apply(perm_simp)
apply(rule refl)
apply(simp add: eqvt_at_def image_eqvt perm_supp_eq)
apply(perm_simp)
apply(rule refl)
done

termination (eqvt) by lexicographic_order

lemma var_pos1:
  assumes "atom y \<notin> supp t"
  shows "var_pos y t = {}"
using assms
apply(induct t rule: var_pos.induct)
apply(simp_all add: lam.supp supp_at_base fresh_at_base)
done

lemma var_pos2:
  shows "var_pos y (Lam [y].t) = {}"
apply(rule var_pos1)
apply(simp add: lam.supp)
done


text {* strange substitution operation *}

nominal_primrec
  subst' :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"  ("_ [_ ::== _]" [90, 90, 90] 90)
where
  "(Var x)[y ::== s] = (if x = y then s else (Var x))"
| "(App t1 t2)[y ::== s] = App (t1[y ::== s]) (t2[y ::== s])"
| "atom x \<sharp> (y, s) \<Longrightarrow> (Lam [x]. t)[y ::== s] = Lam [x].(t[y ::== (App (Var y) s)])"
  apply(simp add: eqvt_def subst'_graph_def)
  apply (rule, perm_simp, rule)
  apply(rule TrueI)
  apply(case_tac x)
  apply(rule_tac y="a" and c="(b, c)" in lam.strong_exhaust)
  apply(auto simp add: fresh_star_def)[3]
  apply(simp_all)
  apply(erule conjE)+
  apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2)
  apply(simp_all add: Abs_fresh_iff)
  apply(simp add: fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
done

termination (eqvt) by lexicographic_order


section {* free name function *}

text {* first returns an atom list *}

nominal_primrec 
  frees_lst :: "lam \<Rightarrow> atom list"
where
  "frees_lst (Var x) = [atom x]"
| "frees_lst (App t1 t2) = frees_lst t1 @ frees_lst t2"
| "frees_lst (Lam [x]. t) = removeAll (atom x) (frees_lst t)"
  unfolding eqvt_def
  unfolding frees_lst_graph_def
  apply (rule, perm_simp, rule)
apply(rule TrueI)
apply(rule_tac y="x" in lam.exhaust)
apply(auto)
apply (erule_tac c="()" in Abs_lst1_fcb2)
apply(simp add: supp_removeAll fresh_def)
apply(simp add: fresh_star_def fresh_Unit)
apply(simp add: eqvt_at_def removeAll_eqvt atom_eqvt)
apply(simp add: eqvt_at_def removeAll_eqvt atom_eqvt)
done

termination (eqvt) by lexicographic_order

text {* a small test lemma *}
lemma shows "supp t = set (frees_lst t)"
  by (induct t rule: frees_lst.induct) (simp_all add: lam.supp supp_at_base)

text {* second returns an atom set - therefore needs an invariant *}

nominal_primrec (invariant "\<lambda>x (y::atom set). finite y")
  frees_set :: "lam \<Rightarrow> atom set"
where
  "frees_set (Var x) = {atom x}"
| "frees_set (App t1 t2) = frees_set t1 \<union> frees_set t2"
| "frees_set (Lam [x]. t) = (frees_set t) - {atom x}"
  apply(simp add: eqvt_def frees_set_graph_def)
  apply(rule, perm_simp, rule)
  apply(erule frees_set_graph.induct)
  apply(auto)[9]
  apply(rule_tac y="x" in lam.exhaust)
  apply(auto)[3]
  apply(simp)
  apply(erule_tac c="()" in Abs_lst1_fcb2)
  apply(simp add: fresh_minus_atom_set)
  apply(simp add: fresh_star_def fresh_Unit)
  apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)
  apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)
  done

termination (eqvt) 
  by lexicographic_order

lemma "frees_set t = supp t"
  by (induct rule: frees_set.induct) (simp_all add: lam.supp supp_at_base)

section {* height function *}

nominal_primrec
  height :: "lam \<Rightarrow> int"
where
  "height (Var x) = 1"
| "height (App t1 t2) = max (height t1) (height t2) + 1"
| "height (Lam [x].t) = height t + 1"
  apply(simp add: eqvt_def height_graph_def)
  apply (rule, perm_simp, rule)
  apply(rule TrueI)
  apply(rule_tac y="x" in lam.exhaust)
  apply(auto)
  apply (erule_tac c="()" in Abs_lst1_fcb2)
  apply(simp_all add: fresh_def pure_supp eqvt_at_def fresh_star_def)
  done

termination (eqvt)
  by lexicographic_order
  
thm height.simps

  
section {* capture-avoiding substitution *}

nominal_primrec
  subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"  ("_ [_ ::= _]" [90, 90, 90] 90)
where
  "(Var x)[y ::= s] = (if x = y then s else (Var x))"
| "(App t1 t2)[y ::= s] = App (t1[y ::= s]) (t2[y ::= s])"
| "atom x \<sharp> (y, s) \<Longrightarrow> (Lam [x]. t)[y ::= s] = Lam [x].(t[y ::= s])"
  unfolding eqvt_def subst_graph_def
  apply (rule, perm_simp, rule)
  apply(rule TrueI)
  apply(auto simp add: lam.distinct lam.eq_iff)
  apply(rule_tac y="a" and c="(aa, b)" in lam.strong_exhaust)
  apply(blast)+
  apply(simp_all add: fresh_star_def fresh_Pair_elim)
  apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2)
  apply(simp_all add: Abs_fresh_iff)
  apply(simp add: fresh_star_def fresh_Pair)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
done

termination (eqvt)
  by lexicographic_order

thm subst.eqvt

lemma forget:
  shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"
  by (nominal_induct t avoiding: x s rule: lam.strong_induct)
     (auto simp add: lam.fresh fresh_at_base)

text {* same lemma but with subst.induction *}
lemma forget2:
  shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"
  by (induct t x s rule: subst.induct)
     (auto simp add: lam.fresh fresh_at_base fresh_Pair)

lemma fresh_fact:
  fixes z::"name"
  assumes a: "atom z \<sharp> s"
      and b: "z = y \<or> atom z \<sharp> t"
  shows "atom z \<sharp> t[y ::= s]"
  using a b
  by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
      (auto simp add: lam.fresh fresh_at_base)

lemma substitution_lemma:  
  assumes a: "x \<noteq> y" "atom x \<sharp> u"
  shows "t[x ::= s][y ::= u] = t[y ::= u][x ::= s[y ::= u]]"
using a 
by (nominal_induct t avoiding: x y s u rule: lam.strong_induct)
   (auto simp add: fresh_fact forget)

lemma subst_rename: 
  assumes a: "atom y \<sharp> t"
  shows "t[x ::= s] = ((y \<leftrightarrow> x) \<bullet>t)[y ::= s]"
using a 
apply (nominal_induct t avoiding: x y s rule: lam.strong_induct)
apply (auto simp add: lam.fresh fresh_at_base)
done

lemma height_ge_one:
  shows "1 \<le> (height e)"
by (induct e rule: lam.induct) (simp_all)

theorem height_subst:
  shows "height (e[x::=e']) \<le> ((height e) - 1) + (height e')"
proof (nominal_induct e avoiding: x e' rule: lam.strong_induct)
  case (Var y)
  have "1 \<le> height e'" by (rule height_ge_one)
  then show "height (Var y[x::=e']) \<le> height (Var y) - 1 + height e'" by simp
next
  case (Lam y e1)
  hence ih: "height (e1[x::=e']) \<le> ((height e1) - 1) + (height e')" by simp
  moreover
  have vc: "atom y\<sharp>x" "atom y\<sharp>e'" by fact+ (* usual variable convention *)
  ultimately show "height ((Lam [y]. e1)[x::=e']) \<le> height (Lam [y]. e1) - 1 + height e'" by simp
next
  case (App e1 e2)
  hence ih1: "height (e1[x::=e']) \<le> ((height e1) - 1) + (height e')"
    and ih2: "height (e2[x::=e']) \<le> ((height e2) - 1) + (height e')" by simp_all
  then show "height ((App e1 e2)[x::=e']) \<le> height (App e1 e2) - 1 + height e'"  by simp
qed

subsection {* single-step beta-reduction *}

inductive 
  beta :: "lam \<Rightarrow> lam \<Rightarrow> bool" (" _ \<longrightarrow>b _" [80,80] 80)
where
  b1[intro]: "t1 \<longrightarrow>b t2 \<Longrightarrow> App t1 s \<longrightarrow>b App t2 s"
| b2[intro]: "s1 \<longrightarrow>b s2 \<Longrightarrow> App t s1 \<longrightarrow>b App t s2"
| b3[intro]: "t1 \<longrightarrow>b t2 \<Longrightarrow> Lam [x]. t1 \<longrightarrow>b Lam [x]. t2"
| b4[intro]: "atom x \<sharp> s \<Longrightarrow> App (Lam [x]. t) s \<longrightarrow>b t[x ::= s]"

equivariance beta

nominal_inductive beta
  avoids b4: "x"
  by (simp_all add: fresh_star_def fresh_Pair lam.fresh fresh_fact)

text {* One-Reduction *}

inductive 
  One :: "lam \<Rightarrow> lam \<Rightarrow> bool" (" _ \<longrightarrow>1 _" [80,80] 80)
where
  o1[intro]: "Var x \<longrightarrow>1 Var x"
| o2[intro]: "\<lbrakk>t1 \<longrightarrow>1 t2; s1 \<longrightarrow>1 s2\<rbrakk> \<Longrightarrow> App t1 s1 \<longrightarrow>1 App t2 s2"
| o3[intro]: "t1 \<longrightarrow>1 t2 \<Longrightarrow> Lam [x].t1 \<longrightarrow>1 Lam [x].t2"
| o4[intro]: "\<lbrakk>atom x \<sharp> (s1, s2); t1 \<longrightarrow>1 t2; s1 \<longrightarrow>1 s2\<rbrakk> \<Longrightarrow> App (Lam [x].t1) s1 \<longrightarrow>1 t2[x ::= s2]"

equivariance One

nominal_inductive One 
  avoids o3: "x"
      |  o4: "x"
  by (simp_all add: fresh_star_def fresh_Pair lam.fresh fresh_fact)

lemma One_refl:
  shows "t \<longrightarrow>1 t"
by (nominal_induct t rule: lam.strong_induct) (auto)

lemma One_subst: 
  assumes a: "t1 \<longrightarrow>1 t2" "s1 \<longrightarrow>1 s2"
  shows "t1[x ::= s1] \<longrightarrow>1 t2[x ::= s2]" 
using a 
apply(nominal_induct t1 t2 avoiding: s1 s2 x rule: One.strong_induct)
apply(auto simp add: substitution_lemma fresh_at_base fresh_fact fresh_Pair)
done

lemma better_o4_intro:
  assumes a: "t1 \<longrightarrow>1 t2" "s1 \<longrightarrow>1 s2"
  shows "App (Lam [x]. t1) s1 \<longrightarrow>1 t2[ x ::= s2]"
proof -
  obtain y::"name" where fs: "atom y \<sharp> (x, t1, s1, t2, s2)" by (rule obtain_fresh)
  have "App (Lam [x]. t1) s1 = App (Lam [y]. ((y \<leftrightarrow> x) \<bullet> t1)) s1" using fs
    by (auto simp add: lam.eq_iff Abs1_eq_iff' flip_def fresh_Pair fresh_at_base)
  also have "\<dots> \<longrightarrow>1 ((y \<leftrightarrow> x) \<bullet> t2)[y ::= s2]" using fs a by (auto simp add: One.eqvt)
  also have "\<dots> = t2[x ::= s2]" using fs by (simp add: subst_rename[symmetric])
  finally show "App (Lam [x].t1) s1 \<longrightarrow>1 t2[x ::= s2]" by simp
qed

section {* Locally Nameless Terms *}

nominal_datatype ln = 
  LNBnd nat
| LNVar name
| LNApp ln ln
| LNLam ln

fun
  lookup :: "name list \<Rightarrow> nat \<Rightarrow> name \<Rightarrow> ln" 
where
  "lookup [] n x = LNVar x"
| "lookup (y # ys) n x = (if x = y then LNBnd n else (lookup ys (n + 1) x))"

lemma supp_lookup:
  shows "supp (lookup xs n x) \<subseteq> {atom x}"
  apply(induct arbitrary: n rule: lookup.induct)
  apply(simp add: ln.supp supp_at_base)
  apply(simp add: ln.supp pure_supp)
  done

lemma supp_lookup_in:
  shows "x \<in> set xs \<Longrightarrow> supp (lookup xs n x) = {}"
  by (induct arbitrary: n rule: lookup.induct)(auto simp add: ln.supp pure_supp)

lemma supp_lookup_notin:
  shows "x \<notin> set xs \<Longrightarrow> supp (lookup xs n x) = {atom x}"
  by (induct arbitrary: n rule: lookup.induct) (auto simp add: ln.supp pure_supp supp_at_base)

lemma supp_lookup_fresh:
  shows "atom ` set xs \<sharp>* lookup xs n x"
  by (case_tac "x \<in> set xs") (auto simp add: fresh_star_def fresh_def supp_lookup_in supp_lookup_notin)

lemma lookup_eqvt[eqvt]:
  shows "(p \<bullet> lookup xs n x) = lookup (p \<bullet> xs) (p \<bullet> n) (p \<bullet> x)"
  by (induct xs arbitrary: n) (simp_all add: permute_pure)

text {* Function that translates lambda-terms into locally nameless terms *}

nominal_primrec (invariant "\<lambda>(_, xs) y. atom ` set xs \<sharp>* y")
  trans :: "lam \<Rightarrow> name list \<Rightarrow> ln"
where
  "trans (Var x) xs = lookup xs 0 x"
| "trans (App t1 t2) xs = LNApp (trans t1 xs) (trans t2 xs)"
| "atom x \<sharp> xs \<Longrightarrow> trans (Lam [x]. t) xs = LNLam (trans t (x # xs))"
  apply (simp add: eqvt_def trans_graph_def)
  apply (rule, perm_simp, rule)
  apply (erule trans_graph.induct)
  apply (auto simp add: ln.fresh)[3]
  apply (simp add: supp_lookup_fresh)
  apply (simp add: fresh_star_def ln.fresh)
  apply (simp add: ln.fresh fresh_star_def)
  apply(auto)[1]
  apply (rule_tac y="a" and c="b" in lam.strong_exhaust)
  apply (auto simp add: fresh_star_def)[3]
  apply(simp_all)
  apply(erule conjE)+
  apply (erule_tac c="xsa" in Abs_lst1_fcb2')
  apply (simp add: fresh_star_def)
  apply (simp add: fresh_star_def)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  done

termination (eqvt)
  by lexicographic_order


text {* count the occurences of lambdas in a term *}

nominal_primrec
  cntlams :: "lam  \<Rightarrow> nat"
where
  "cntlams (Var x) = 0"
| "cntlams (App t1 t2) = (cntlams t1) + (cntlams t2)"
| "cntlams (Lam [x]. t) = Suc (cntlams t)"
  apply(simp add: eqvt_def cntlams_graph_def)
  apply(rule, perm_simp, rule)
  apply(rule TrueI)
  apply(rule_tac y="x" in lam.exhaust)
  apply(auto)[3]
  apply(all_trivials)
  apply(simp)
  apply(simp)
  apply(erule_tac c="()" in Abs_lst1_fcb2')
  apply(simp add: pure_fresh)
  apply(simp add: fresh_star_def pure_fresh)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  done

termination (eqvt)
  by lexicographic_order


text {* count the bound-variable occurences in a lambda-term *}

nominal_primrec
  cntbvs :: "lam \<Rightarrow> name list \<Rightarrow> nat"
where
  "cntbvs (Var x) xs = (if x \<in> set xs then 1 else 0)"
| "cntbvs (App t1 t2) xs = (cntbvs t1 xs) + (cntbvs t2 xs)"
| "atom x \<sharp> xs \<Longrightarrow> cntbvs (Lam [x]. t) xs = cntbvs t (x # xs)"
  apply(simp add: eqvt_def cntbvs_graph_def)
  apply(rule, perm_simp, rule)
  apply(rule TrueI)
  apply(case_tac x)
  apply(rule_tac y="a" and c="b" in lam.strong_exhaust)
  apply(auto simp add: fresh_star_def)[3]
  apply(all_trivials)
  apply(simp)
  apply(simp)
  apply(simp)
  apply(erule conjE)
  apply(erule Abs_lst1_fcb2')
  apply(simp add: pure_fresh fresh_star_def)
  apply(simp add: fresh_star_def)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)
  done

termination (eqvt)
  by lexicographic_order

section {* De Bruijn Terms *}

nominal_datatype db = 
  DBVar nat
| DBApp db db
| DBLam db

instance db :: pure
  apply default
  apply (induct_tac x rule: db.induct)
  apply (simp_all add: permute_pure)
  done

lemma fresh_at_list: "atom x \<sharp> xs \<longleftrightarrow> x \<notin> set xs"
  unfolding fresh_def supp_set[symmetric]
  by (induct xs) (auto simp add: supp_of_finite_insert supp_at_base supp_set_empty)

fun
  vindex :: "name list \<Rightarrow> name \<Rightarrow> nat \<Rightarrow> db option" 
where
  "vindex [] v n = None"
| "vindex (h # t) v n = (if v = h then (Some (DBVar n)) else (vindex t v (Suc n)))"

lemma vindex_eqvt[eqvt]:
  "(p \<bullet> vindex l v n) = vindex (p \<bullet> l) (p \<bullet> v) (p \<bullet> n)"
  by (induct l arbitrary: n) (simp_all add: permute_pure)

nominal_primrec
  transdb :: "lam \<Rightarrow> name list \<Rightarrow> db option"
where
  "transdb (Var x) l = vindex l x 0"
| "transdb (App t1 t2) xs = 
      Option.bind (transdb t1 xs) (\<lambda>d1. Option.bind (transdb t2 xs) (\<lambda>d2. Some (DBApp d1 d2)))"
| "x \<notin> set xs \<Longrightarrow> transdb (Lam [x].t) xs = Option.map DBLam (transdb t (x # xs))"
  unfolding eqvt_def transdb_graph_def
  apply (rule, perm_simp, rule)
  apply(rule TrueI)
  apply (case_tac x)
  apply (rule_tac y="a" and c="b" in lam.strong_exhaust)
  apply (auto simp add: fresh_star_def fresh_at_list)[3]
  apply(simp_all)
  apply(elim conjE)
  apply (erule_tac c="xsa" in Abs_lst1_fcb2')
  apply (simp add: pure_fresh)
  apply(simp add: fresh_star_def fresh_at_list)
  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq eqvts eqvts_raw)+
  done

termination (eqvt)
  by lexicographic_order

lemma transdb_eqvt[eqvt]:
  "p \<bullet> transdb t l = transdb (p \<bullet>t) (p \<bullet>l)"
  apply (nominal_induct t avoiding: l rule: lam.strong_induct)
  apply (simp add: vindex_eqvt)
  apply (simp_all add: permute_pure)
  apply (simp add: fresh_at_list)
  apply (subst transdb.simps)
  apply (simp add: fresh_at_list[symmetric])
  apply (drule_tac x="name # l" in meta_spec)
  apply auto
  done

lemma db_trans_test:
  assumes a: "y \<noteq> x"
  shows "transdb (Lam [x]. Lam [y]. App (Var x) (Var y)) [] = 
  Some (DBLam (DBLam (DBApp (DBVar 1) (DBVar 0))))"
  using a by simp

lemma supp_subst:
  shows "supp (t[x ::= s]) \<subseteq> (supp t - {atom x}) \<union> supp s"
  by (induct t x s rule: subst.induct) (auto simp add: lam.supp supp_at_base)

lemma var_fresh_subst:
  "atom x \<sharp> s \<Longrightarrow> atom x \<sharp> (t[x ::= s])"
  by (induct t x s rule: subst.induct) (auto simp add: lam.supp lam.fresh fresh_at_base)

end