--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal/Ex/Lambda_F_T_FCB2.thy	Sun Jul 03 21:04:46 2011 +0900
@@ -0,0 +1,270 @@
+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