theory Lambda
imports "../Nominal2" 
begin

atom_decl name

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

lemma Abs1_eq_fdest:
  fixes x y :: "'a :: at_base"
    and S T :: "'b :: fs"
  assumes "(Abs_lst [atom x] T) = (Abs_lst [atom y] S)"
  and "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom x \<sharp> f x T"
  and "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom y \<sharp> f x T"
  and "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> T = S \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> (f x T) = f y S"
  and "sort_of (atom x) = sort_of (atom y)"
  shows "f x T = f y S"
using assms apply -
apply (subst (asm) Abs1_eq_iff')
apply simp_all
apply (elim conjE disjE)
apply simp
apply(rule trans)
apply (rule_tac p="(atom x \<rightleftharpoons> atom y)" in supp_perm_eq[symmetric])
apply(rule fresh_star_supp_conv)
apply(simp add: supp_swap fresh_star_def)
apply(simp add: swap_commute)
done

lemma fresh_fun_eqvt_app3:
  assumes a: "eqvt f"
  and b: "a \<sharp> x" "a \<sharp> y" "a \<sharp> z"
  shows "a \<sharp> f x y z"
  using fresh_fun_eqvt_app[OF a b(1)] a b
  by (metis fresh_fun_app)

lemma fresh_fun_eqvt_app4:
  assumes a: "eqvt f"
  and b: "a \<sharp> x" "a \<sharp> y" "a \<sharp> z" "a \<sharp> w"
  shows "a \<sharp> f x y z w"
  using fresh_fun_eqvt_app[OF a b(1)] a b
  by (metis fresh_fun_app)

nominal_primrec
  f
where
  "f f1 f2 f3 (Var x) l = f1 x l"
| "f f1 f2 f3 (App t1 t2) l = f2 t1 t2 (f f1 f2 f3 t1 l) (f f1 f2 f3 t2 l) l"
| "(\<And>t l r. atom x \<sharp> r \<Longrightarrow> atom x \<sharp> f3 x t r l) \<Longrightarrow> (eqvt f1 \<and> eqvt f2 \<and> eqvt f3) \<Longrightarrow> atom x \<sharp> (f1,f2,f3,l) \<Longrightarrow> (f f1 f2 f3 (Lam [x].t) l) = f3 x t (f f1 f2 f3 t (x # l)) l"
  apply (simp add: eqvt_def f_graph_def)
  apply (rule, perm_simp, rule)
  apply(case_tac x)
  apply(rule_tac y="d" and c="z" in lam.strong_exhaust)
  apply(auto simp add: fresh_star_def)
  apply(blast)
  apply blast
  defer
  apply(simp add: fresh_Pair_elim)
  apply(erule Abs1_eq_fdest)
  defer
  apply simp_all
  apply (rule_tac f="f3a" in fresh_fun_eqvt_app4)
  apply assumption
  apply (simp add: fresh_at_base)
  apply assumption
  apply (erule fresh_eqvt_at)
  apply (simp add: supp_Pair supp_fun_eqvt finite_supp)
  apply (simp add: fresh_Pair)
  apply (simp add: fresh_Cons)
  apply (simp add: fresh_Cons fresh_at_base)
  apply (assumption)
  apply (subgoal_tac "\<And>p y r l. p \<bullet> (f3a x y r l) = f3a (p \<bullet> x) (p \<bullet> y) (p \<bullet> r) (p \<bullet> l)")
  apply (subgoal_tac "(atom x \<rightleftharpoons> atom xa) \<bullet> la = la")
  apply (simp add: eqvt_at_def eqvt_def)
  apply (simp add: swap_fresh_fresh)
  apply (simp add: permute_fun_app_eq)
  apply (simp add: eqvt_def)
  prefer 2
  apply (subgoal_tac "atom x \<sharp> f_sumC (f1a, f2a, f3a, t, x # la)")
  apply simp
--"I believe this holds by induction on the graph..."
  unfolding f_sumC_def
  apply (rule_tac y="t" in lam.exhaust)
  apply (subgoal_tac "THE_default undefined (f_graph (f1a, f2a, f3a, t, x # la)) = f1a name (x # la)")
  apply simp
  defer
  apply (rule THE_default1_equality)
  apply simp
  defer
  apply simp
  apply (rule_tac ?f1.0="f1a" in f_graph.intros(1))
  sorry (*this could be defined? *)

termination
  by (relation "measure (\<lambda>(_,_,_,x,_). size x)") (auto simp add: lam.size)

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 [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)

definition
  trans :: "lam \<Rightarrow> name list \<Rightarrow> ln"
where
  "trans t l = f (%x l. lookup l 0 x) (%t1 t2 r1 r2 l. LNApp r1 r2) (%n t r l. LNLam r) t l"

lemma
 "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_all add: trans_def)
apply (subst f.simps)
apply (simp add: ln.fresh)
apply (simp add: eqvt_def)
apply auto
apply (perm_simp, rule)
apply (perm_simp, rule)
apply (perm_simp, rule)
apply (auto simp add: fresh_Pair)[1]
apply (simp_all add: fresh_def supp_def permute_fun_def)[3]
apply (simp add: eqvts permute_pure)
done

lemma lam_strong_exhaust2:
  "\<lbrakk>\<And>name. y = Var name \<Longrightarrow> P; 
    \<And>lam1 lam2. y = App lam1 lam2 \<Longrightarrow> P;
    \<And>name lam. \<lbrakk>{atom name} \<sharp>* c; y = Lam [name]. lam\<rbrakk> \<Longrightarrow> P;
    finite (supp c)\<rbrakk>
    \<Longrightarrow> P"
sorry

nominal_primrec
  g
where
  "(~finite (supp (g1, g2, g3))) \<Longrightarrow> g g1 g2 g3 t = t"
| "finite (supp (g1, g2, g3)) \<Longrightarrow> g g1 g2 g3 (Var x) = g1 x"
| "finite (supp (g1, g2, g3)) \<Longrightarrow> g g1 g2 g3 (App t1 t2) = g2 t1 t2 (g g1 g2 g3 t1) (g g1 g2 g3 t2)"
| "finite (supp (g1, g2, g3)) \<Longrightarrow> atom x \<sharp> (g1,g2,g3) \<Longrightarrow> (g g1 g2 g3 (Lam [x].t)) = g3 x t (g g1 g2 g3 t)"
  apply (simp add: eqvt_def g_graph_def)
  apply (rule, perm_simp, rule)
  apply (case_tac x)
  apply (case_tac "finite (supp (a, b, c))")
  prefer 2
  apply simp
  apply(rule_tac y="d" and c="(a,b,c)" in lam_strong_exhaust2)
  apply simp_all