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