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 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)

lemma the_default_pty:

  assumes f_def: "f == (\<lambda>x::'a. THE_default d (G x))"

  and unique: "\<exists>!y. G x y"

  and pty: "\<And>x y. G x y \<Longrightarrow> P x y"

  shows "P x (f x)"

  apply(subst f_def)

  apply (rule ex1E[OF unique])

  apply (subst THE_default1_equality[OF unique])

  apply assumption

  apply (rule pty)

  apply assumption

  done

lemma Abs_lst1_fcb2:

  fixes a b :: "'a :: at"

    and S T :: "'b :: fs"

    and c::"'c::fs"

  assumes e: "(Abs_lst [atom a] T) = (Abs_lst [atom b] S)"

  and fcb: "\<And>a T. atom a \<sharp> f a T c"

  and fresh: "{atom a, atom b} \<sharp>* c"

  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a T c) = f (p \<bullet> a) (p \<bullet> T) c"

  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b S c) = f (p \<bullet> b) (p \<bullet> S) c"

  shows "f a T c = f b S c"

proof -

  have fin1: "finite (supp (f a T c))"

    apply(rule_tac S="supp (a, T, c)" in supports_finite)

    apply(simp add: supports_def)

    apply(simp add: fresh_def[symmetric])

    apply(clarify)

    apply(subst perm1)

    apply(simp add: supp_swap fresh_star_def)

    apply(simp add: swap_fresh_fresh fresh_Pair)

    apply(simp add: finite_supp)

    done

  have fin2: "finite (supp (f b S c))"

    apply(rule_tac S="supp (b, S, c)" in supports_finite)

    apply(simp add: supports_def)

    apply(simp add: fresh_def[symmetric])

    apply(clarify)

    apply(subst perm2)

    apply(simp add: supp_swap fresh_star_def)

    apply(simp add: swap_fresh_fresh fresh_Pair)

    apply(simp add: finite_supp)

    done

  obtain d::"'a::at" where fr: "atom d \<sharp> (a, b, S, T, c, f a T c, f b S c)" 

    using obtain_fresh'[where x="(a, b, S, T, c, f a T c, f b S c)"]

    apply(auto simp add: finite_supp supp_Pair fin1 fin2)

    done

  have "(a \<leftrightarrow> d) \<bullet> (Abs_lst [atom a] T) = (b \<leftrightarrow> d) \<bullet> (Abs_lst [atom b] S)" 

    apply(simp (no_asm_use) only: flip_def)

    apply(subst swap_fresh_fresh)

    apply(simp add: Abs_fresh_iff)

    using fr

    apply(simp add: Abs_fresh_iff)

    apply(subst swap_fresh_fresh)

    apply(simp add: Abs_fresh_iff)

    using fr

    apply(simp add: Abs_fresh_iff)

    apply(rule e)

    done

  then have "Abs_lst [atom d] ((a \<leftrightarrow> d) \<bullet> T) = Abs_lst [atom d] ((b \<leftrightarrow> d) \<bullet> S)"

    apply (simp add: swap_atom flip_def)

    done

  then have eq: "(a \<leftrightarrow> d) \<bullet> T = (b \<leftrightarrow> d) \<bullet> S"

    by (simp add: Abs1_eq_iff)

  have "f a T c = (a \<leftrightarrow> d) \<bullet> f a T c"

    unfolding flip_def

    apply(rule sym)

    apply(rule swap_fresh_fresh)

    using fcb[where a="a"] 

    apply(simp)

    using fr

    apply(simp add: fresh_Pair)

    done

  also have "... = f d ((a \<leftrightarrow> d) \<bullet> T) c"

    unfolding flip_def

    apply(subst perm1)

    using fresh fr

    apply(simp add: supp_swap fresh_star_def fresh_Pair)

    apply(simp)

    done

  also have "... = f d ((b \<leftrightarrow> d) \<bullet> S) c" using eq by simp

  also have "... = (b \<leftrightarrow> d) \<bullet> f b S c"

    unfolding flip_def

    apply(subst perm2)

    using fresh fr

    apply(simp add: supp_swap fresh_star_def fresh_Pair)

    apply(simp)

    done

  also have "... = f b S c"   

    apply(rule flip_fresh_fresh)

    using fcb[where a="b"] 

    apply(simp)

    using fr

    apply(simp add: fresh_Pair)

    done

  finally show ?thesis by simp

qed

locale test =

  fixes f1::"name \<Rightarrow> name list \<Rightarrow> ('a::pt)"

    and f2::"lam \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> name list \<Rightarrow> ('a::pt)"

    and f3::"name \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> name list \<Rightarrow> ('a::pt)"

  assumes fs: "finite (supp (f1, f2, f3))"

    and eq: "eqvt f1" "eqvt f2" "eqvt f3"

    and fcb1: "\<And>l n. atom ` set l \<sharp>* f1 n l"

    and fcb2: "\<And>l t1 t2 r1 r2. atom ` set l \<sharp>* (r1, r2) \<Longrightarrow> atom ` set l \<sharp>* f2 t1 t2 r1 r2 l"

    and fcb3: "\<And>t l r. atom ` set (x # l) \<sharp>* r \<Longrightarrow> atom ` set (x # l) \<sharp>* f3 x t r l"

begin

nominal_primrec (invariant "\<lambda>(x, l) y. atom ` set l \<sharp>* y")

  f

where

  "f (Var x) l = f1 x l"

| "f (App t1 t2) l = f2 t1 t2 (f t1 l) (f t2 l) l"

| "atom x \<sharp> l \<Longrightarrow> (f (Lam [x].t) l) = f3 x t (f t (x # l)) l"

  apply (simp add: eqvt_def f_graph_def)

  apply (rule, perm_simp)

  apply (simp only: eq[unfolded eqvt_def])

  apply (erule f_graph.induct)

  apply (simp add: fcb1)

  apply (simp add: fcb2 fresh_star_Pair)

  apply simp

  apply (subgoal_tac "atom ` set (xa # l) \<sharp>* f3 xa t (f_sum (t, xa # l)) l")

  apply (simp add: fresh_star_def)

  apply (rule fcb3)

  apply (simp add: fresh_star_def fresh_def)

  apply simp_all

  apply(case_tac x)

  apply(rule_tac y="a" and c="b" in lam.strong_exhaust)

  apply(auto simp add: fresh_star_def)

  apply(erule_tac Abs_lst1_fcb2)

--"?"

  apply (subgoal_tac "atom ` set (a # la) \<sharp>* f3 a T (f_sumC (T, a # la)) la")

  apply (simp add: fresh_star_def)

  apply (rule fcb3)

  apply (simp add: fresh_star_def)

  apply (rule fresh_fun_eqvt_app4[OF eq(3)])

  apply (simp add: fresh_at_base)

  apply assumption

  apply (erule fresh_eqvt_at)

  apply (simp add: finite_supp)

  apply (simp add: fresh_Pair fresh_Cons fresh_at_base)

  apply assumption

  apply (subgoal_tac "\<And>p y r l. p \<bullet> (f3 x y r l) = f3 (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)

  apply (simp add: swap_fresh_fresh)

  apply (simp add: permute_fun_app_eq eq[unfolded eqvt_def])

  apply simp

  done

termination

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

end

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 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)

lemma fresh_at_list: "atom x \<sharp> xs \<longleftrightarrow> x \<notin> set xs"

  unfolding fresh_def supp_set[symmetric]

  apply (induct xs)

  apply (simp add: supp_set_empty)

  apply simp

  apply auto

  apply (simp_all add: insert_absorb UnI2 finite_set supp_of_finite_insert supp_at_base)

  done

interpretation trans: test

  "%x l. lookup l 0 x"

  "%t1 t2 r1 r2 l. LNApp r1 r2"

  "%n t r l. LNLam r"

  apply default

  apply (auto simp add: pure_fresh supp_Pair)

  apply (simp_all add: fresh_def supp_def permute_fun_def permute_pure lookup_eqvt)[3]

  apply (simp_all add: eqvt_def permute_fun_def permute_pure lookup_eqvt)

  apply (simp add: fresh_star_def)

  apply (rule_tac x="0 :: nat" in spec)

  apply (induct_tac l)

  apply (simp add: ln.fresh pure_fresh)

  apply (auto simp add: ln.fresh pure_fresh)[1]

  apply (case_tac "a \<in> set list")

  apply simp

  apply (rule_tac f="lookup" in fresh_fun_eqvt_app3)

  unfolding eqvt_def

  apply rule

  using eqvts_raw(35)

  apply auto[1]

  apply (simp add: fresh_at_list)

  apply (simp add: pure_fresh)

  apply (simp add: fresh_at_base)

  apply (simp add: fresh_star_Pair fresh_star_def ln.fresh)

  apply (simp add: fresh_star_def ln.fresh)

  done

thm trans.f.simps

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 simp_all

  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 auto

  apply blast

  apply (simp add: Abs1_eq_iff fresh_star_def)

  sorry

termination

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

end