theory Lambda_add
imports "Lambda"
begin

nominal_primrec
  addlam :: "lam \<Rightarrow> lam"
where
  "fv \<noteq> x \<Longrightarrow> addlam (Var x) = Lam [fv]. App (Var fv) (Var x)"
| "addlam (App t1 t2) = App t1 t2"
| "addlam (Lam [x]. t) = Lam [x].t"
oops


nominal_primrec
  addlam_pre :: "lam \<Rightarrow> lam \<Rightarrow> lam"
where
  "fv \<noteq> x \<Longrightarrow> addlam_pre (Var x) (Lam [fv].it) = Lam [fv]. App (Var fv) (Var x)"
| "addlam_pre (Var x) (App t1 t2) = Var x"
| "addlam_pre (Var x) (Var v) = Var x"
| "addlam_pre (App t1 t2) it = App t1 t2"
| "addlam_pre (Lam [x]. t) it = Lam [x].t"
apply (subgoal_tac "\<And>p a x. addlam_pre_graph a x \<Longrightarrow> addlam_pre_graph (p \<bullet> a) (p \<bullet> x)")
apply (simp add: eqvt_def)
apply (intro allI)
apply(simp add: permute_fun_def)
apply(rule ext)
apply(rule ext)
apply(simp add: permute_bool_def)
apply rule
apply(drule_tac x="p" in meta_spec)
apply(drule_tac x="- p \<bullet> x" in meta_spec)
apply(drule_tac x="- p \<bullet> xa" in meta_spec)
apply simp
apply(drule_tac x="-p" in meta_spec)
apply(drule_tac x="x" in meta_spec)
apply(drule_tac x="xa" in meta_spec)
apply simp
apply (erule addlam_pre_graph.induct)
apply (simp_all add: addlam_pre_graph.intros)
apply (simp_all add:lam.distinct)
apply clarify
apply simp
apply (rule_tac y="a" in lam.exhaust)
apply(auto simp add: lam.distinct lam.eq_iff)
prefer 2 apply blast
prefer 2 apply blast
apply (rule_tac y="b" and c="name" in lam.strong_exhaust)
apply(auto simp add: lam.distinct lam.eq_iff)[3]
apply blast
apply(drule_tac x="namea" in meta_spec)
apply(drule_tac x="name" in meta_spec)
apply(drule_tac x="lam" in meta_spec)
apply (auto simp add: fresh_star_def atom_name_def fresh_at_base)[1]
apply (auto simp add: lam.eq_iff Abs1_eq_iff fresh_def lam.supp supp_at_base)
done

termination
  by (relation "measure (\<lambda>(t,_). size t)")
     (simp_all add: lam.size)

function
  addlam :: "lam \<Rightarrow> lam"
where
  "addlam (Var x) = addlam_pre (Var x) (Lam [x]. (Var x))"
| "addlam (App t1 t2) = App t1 t2"
| "addlam (Lam [x]. t) = Lam [x].t"
apply (simp_all add:lam.distinct)
apply (rule_tac y="x" in lam.exhaust)
apply auto
apply (simp add:lam.eq_iff)
done

termination
  by (relation "measure (\<lambda>(t). size t)")
     (simp_all add: lam.size)

lemma addlam':
  "fv \<noteq> x \<Longrightarrow> addlam (Var x) = Lam [fv]. App (Var fv) (Var x)"
  apply simp
  apply (subgoal_tac "Lam [x]. Var x = Lam [fv]. Var fv")
  apply simp
  apply (simp add: lam.eq_iff Abs1_eq_iff lam.supp fresh_def supp_at_base)
  done

nominal_primrec
  Pair_pre
where
  "atom z \<sharp> (M, N) \<Longrightarrow> (Pair_pre M N) (Lam [z]. _) = (Lam [z]. App (App (Var z) M) N)"
| "(Pair_pre M N) (App l r) = (App l r)"
| "(Pair_pre M N) (Var v) = (Var v)"
apply (subgoal_tac "\<And>p a x. Pair_pre_graph a x \<Longrightarrow> Pair_pre_graph (p \<bullet> a) (p \<bullet> x)")
apply (simp add: eqvt_def)
apply (intro allI)
apply(simp add: permute_fun_def)
apply(rule ext)
apply(rule ext)
apply(simp add: permute_bool_def)
apply rule
apply(drule_tac x="p" in meta_spec)
apply(drule_tac x="- p \<bullet> x" in meta_spec)
apply(drule_tac x="- p \<bullet> xa" in meta_spec)
apply simp
apply(drule_tac x="-p" in meta_spec)
apply(drule_tac x="x" in meta_spec)
apply(drule_tac x="xa" in meta_spec)
apply simp
apply (erule Pair_pre_graph.induct)
apply (simp_all add: Pair_pre_graph.intros)[3]
  apply (case_tac x)
  apply clarify
  apply (simp add: symmetric[OF atomize_conjL])
  apply (rule_tac y="c" and c="(ab, ba)" in lam.strong_exhaust)
  apply (simp_all add: lam.distinct symmetric[OF atomize_conjL] lam.eq_iff Abs1_eq_iff fresh_star_def)
  apply auto[1]
  apply (simp_all add: fresh_Pair lam.fresh fresh_at_base)
  apply (rule swap_fresh_fresh[symmetric])
  prefer 3
  apply (rule swap_fresh_fresh[symmetric])
  apply simp_all
  done

termination
  by (relation "measure (\<lambda>(t,_). size t)")
     (simp_all add: lam.size)

consts cx :: name

fun
  Pair where
  "(Pair M N) = (Pair_pre M N (Lam [cx]. Var cx))"

lemma Pair : "atom z \<sharp> (M, N) \<Longrightarrow> Pair M N = (Lam [z]. App (App (Var z) M) N)"
  apply simp
  apply (subgoal_tac "Lam [cx]. Var cx = Lam[z]. Var z")
  apply simp
  apply (auto simp add: lam.eq_iff Abs1_eq_iff)
  by (metis Rep_name_inject atom_name_def fresh_at_base lam.fresh(1))


definition Lam_I_pre : "\<II> \<equiv> Lam[cx]. Var cx"

lemma Lam_I : "\<II> = Lam[x]. Var x"
  by (simp add: Lam_I_pre lam.eq_iff Abs1_eq_iff lam.fresh supp_at_base)
     (metis Rep_name_inverse atom_name_def fresh_at_base)

definition c1 :: name where "c1 \<equiv> Abs_name (Atom (Sort ''Lambda.name'' []) 1)"
definition c2 :: name where "c2 \<equiv> Abs_name (Atom (Sort ''Lambda.name'' []) 2)"

lemma c1_c2[simp]:
  "c1 \<noteq> c2"
  unfolding c1_def c2_def
  by (simp add: Abs_name_inject)

definition Lam_F_pre : "\<FF> \<equiv> Lam[c1]. Lam[c2]. Var c2"
lemma Lam_F : "\<FF> = Lam[x]. Lam[y]. Var y"
  by (simp add: Lam_F_pre lam.eq_iff Abs1_eq_iff lam.fresh supp_at_base)
     (metis Rep_name_inverse atom_name_def fresh_at_base)

end